Prove a Continuous Periodic Function is Bounded and Uniformly Continuous

Realizability: An Introduction to its Categorical Side

In Studies in Logic and the Foundations of Mathematics, 2008

Theorem 3.3.10

i)

There is a uniformly continuous function defined on [0,1] such that for all x ∈ [0, 1] f(x) > 0, yet inf _{x ∈ [0, 1]} f(x) = 0.

ii)

There is a continuous function f on [0, 1] which is unbounded, and therefore not uniformly continuous.

Proof

(i) Let (I_n ) _n be a covering of [0, 1] by open intervals (r_n, s_n ) which has the property of the covering in theorem 3.3.7, that for every n, ${\Sigma }_{k=0}^n\left(s_k-r_k\right)<\frac{1}{2}$ say For every n let f_n be defined by

$f_n\left(x\right)=max\left\{0,\frac{1}{2}\left(s_n-r_n\right)-\left|x-\frac{1}{2}\left(r_n+s_n\right)\right|\right\}$

Define f by $f\left(x\right)=$ ${\Sigma }_{n=0}^{\infty }{2}^{-n}{f}_n\left(x\right).$ .

Then f is defined everywhere, and is uniformly continuous: if |x – y| < δ then |f_n (x) –f_n (y)| < δ for all n, so |f(x) –f(y)| < 2δ. Furthermore, for every x ∈ I_n, f_n (x) > 0 so since (I_n ) _n covers, f(x) > 0 everywhere. But by the property of this cover, it follows that no finite part of it covers [0, 1]; if x ∉ I ₀ ∪ · · · ∪ I_n , then f(x) ≤ 2⁻ⁿ . This means that inf _{x ∈ [0, 1]} f(x) = 0, as claimed.

(ii) Take the function f from i), and define g by g(x) =f(x)⁻¹.

For material on differential equations in ɛff, see [141].

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Space C(E,E′)

Agamirza Bashirov , in Mathematical Analysis Fundamentals, 2014

Exercises

6.1

Prove that the composition of two uniformly continuous functions is uniformly continuous.

6.2

Show that the functions

(a)

$f(x)=1/(1+x^2)\text{,}x\in \mathbb{R}$ ,

(b)

$g(x)=x/(1+x^2)\text{,}x\in \mathbb{R}$ ,

are uniformly continuous, while the functions

(c)

$f(x)=x^2\text{,}x\in \mathbb{R}$ ,

(d)

$g(x)=\sqrt{\mid x\mid }\text{,}x\in \mathbb{R}$ ,

are not.

6.3

Let $(E\text{,}d)$ and $(E^{\prime }\text{,}{d}^{\prime })$ be metric spaces. A function $f:E\to E^{\prime }$ is said to be Lipschitz ⁶² continuous or satisfy the Lipschitz condition if there exists $k\in \mathbb{R}$ such that $d^{\prime }(f(p)\text{,}f(q))\le \mathit{kd}(p\text{,}q)$ for every $p\text{,}q\in E$ . Prove that Lipschitz continuous functions are uniformly continuous.

6.4

Let $(E\text{,}d)$ and $(E^{\prime }\text{,}{d}^{\prime })$ be metric spaces. A function $f:E\to E^{\prime }$ is said to be $\alpha$ -Hölder ⁶³ continuous or satisfy the $\alpha$ -Hölder condition with $0<\alpha <1$ if there exists $k\in \mathbb{R}$ such that $d^{\prime }(f(p)\text{,}f(q))\le \mathit{kd}{(p\text{,}q)}^{\alpha }$ for every $p\text{,}q\in E$ . Prove that $\alpha$ -Hölder continuous functions are uniformly continuous.

6.5

Prove that the sum and difference of two uniformly continuous functions is uniformly continuous. Give an example of two uniformly continuous functions such that their product is not uniformly continuous.

6.6

Show that if $f$ is continuous on $(0\text{,}1)$ but unbounded there, then $f$ is not uniformly continuous.

6.7

Let $E$ and $E^{\prime }$ be metric spaces and let $f:E\to E^{\prime }$ be uniformly continuous. Prove that if $\{p_n\}$ is Cauchy in $E$ , then $\{f(p_n)\}$ is Cauchy in $E^{\prime }$ . Give an example of a continuous function $g:E\to E^{\prime }$ such that $\{p_n\}$ is Cauchy in $E$ while $\{g(p_n)\}$ is not Cauchy in $E^{\prime }$ .

6.8

Let $E$ be a compact metric space and let $E^{\prime }$ be a metric space. For $p_0\in E$ , define the function $F:C(E\text{,}{E}^{\prime })\to E^{\prime }$ by $F(f)=f(p_0)\text{,}f\in C(E.E^{\prime })$ . Show that $F$ is uniformly continuous.

6.9

Let $E$ be a compact metric space and let $f\text{,}g\in C(E)$ . Define the function $h:\mathbb{R}\to C(E)$ by $h(x)=\mathit{xf}+(1-x)g\text{,}x\in \mathbb{R}$ . Show that $h$ is uniformly continuous.

6.10

Let $f:\mathbb{R}\to \mathbb{R}$ be uniformly continuous and let $a\in \mathbb{R}$ . Define the function $g:\mathbb{R}\to \mathbb{R}$ by $g(x)=f(x+a)\text{,}x\in \mathbb{R}$ . Show that $g$ is uniformly continuous.

6.11

Let $f$ have a continuous inverse $f^{-1}$ . If $f$ is uniformly continuous, can we conclude that $f^{-1}$ is also uniformly continuous?

6.12

Show that the sequence of functions $f_n(x)=x^n(1-x)\text{,}0\le x\le 1\text{,}n=1\text{,}2\text{,}\dots \text{,}$ converges uniformly on $[0\text{,}1]$ .

Hint: At first, show that the sequence of functions $g_n(x)=x^n\text{,}0\le x\le a\text{,}n=1\text{,}2\text{,}\dots \text{,}$ is uniformly convergent for every $0<a<1$ .

6.13

Determine whether the sequence of functions $f_n$ is uniformly convergent on $[0\text{,}1]$ if

(a)

$f_n(x)=\frac{x}{1+{\mathit{nx}}^2}$ ;

(b)

$f_n(x)=\frac{\mathit{nx}}{1+{\mathit{nx}}^2}$ ;

(c)

$f_n(x)=\frac{\mathit{nx}}{1+n^2{x}^2}$ .

6.14

Let $f:\mathbb{R}\to \mathbb{R}$ be uniformly continuous and let the numerical sequence $\{a_n\}$ be convergent to zero. Define $f_n(x)=f(x+a_n)\text{,}x\in \mathbb{R}\text{,}n=1\text{,}2\text{,}\dots$ Show that $\{f_n\}$ converges uniformly.

6.15

Show that the closed ball in $C(0\text{,}1)$ , centered at the zero function with the radius 1, is not compact.

Hint: Use the sequence of functions $f_n(x)=x^n\text{,}0\le x\le 1\text{,}n=1\text{,}2\text{,}\dots$

6.16

Let $E$ be a metric space and let $f_n:E\to \mathbb{R}$ be uniformly convergent. Show that if $f:E\to \mathbb{R}$ is bounded, then $f_{n}f$ is uniformly convergent.

6.17

Let $E$ be a metric space and let $f_n:E\to \mathbb{R}$ be uniformly convergent. Show that if $\mid f_n(p)\mid \ge c>0$ for every $p\in E$ and $n=1\text{,}2\text{,}\dots \text{,}$ then $1/f_n$ converges uniformly.

6.18

Let $E$ be a compact metric space and let $\{f_n\}$ be a sequence in $C(E)$ , satisfying $f_1(p)\le f_2(p)\le \cdots$ for every $p\in E$ . Show that if $\{f_n\}$ converges to $f\in C(E)$ , then this convergence is uniform.

6.19

A function of the form

$f(x_1\text{,}\dots \text{,}{x}_k)={\displaystyle \sum _{n_1+\cdots +n_k=n}}{a}_{n_1\text{,}\dots \text{,}{n}_k}{x}_1^{n_1}\cdots x_k^{n_k}\text{,}$

where $a_{n_1\text{,}\dots \text{,}{n}_k}\in \mathbb{R}\text{,}n$ is a nonnegative integer, and the summation is taken over all $k$ -tuples of nonnegative integers $n_1\text{,}\dots \text{,}{n}_k$ with $n_1+\cdots +n_k=n$ , is called a polynomial in $x_1\text{,}\dots \text{,}{x}_k$ . Show that if $E$ is a compact subset of ${\mathbb{R}}^k$ then the polynomials in $x_1\text{,}\dots \text{,}{x}_k$ are dense in $C(E)$ .

6.20

Show that if $E$ is a compact subset of ${\mathbb{R}}^m$ then $C(E\text{,}{\mathbb{R}}^k)$ is separable.

6.21

Show that the set $P$ of all polynomials on $[0\text{,}1]$ forms an algebra in $C(0\text{,}1)$ , which separates points in $[0\text{,}1]$ and $\mathbb{R}$ . However, show that $P$ is not a subset of the type $\mathcal{S}$ as described in Theorem 6.23. Conclude that Theorem 6.23 cannot be applied to the set $P$ while Theorem 6.25 can be applied to $P$ .

6.22

A function $f:[a\text{,}b]\to \mathbb{R}$ is said to be piecewise linear if there are the numbers $a=t_0<\cdots <t_n=b$ such that on each of the closed intervals $[t_{i+1}\text{,}{t}_i]\text{,}f(x)=a_{i}x+b_i$ for some $a_i\text{,}{b}_i\in \mathbb{R}\text{,}i=0\text{,}1\text{,}\dots \text{,}n-1$ . Let $\mathcal{S}$ be the set of all piecewise linear functions on $[a\text{,}b]$ . Show that $\mathcal{S}$ satisfies the conditions of Theorem 6.23.

6.23

Let $(E\text{,}d)$ be a compact metric space and let $(E^{\prime }\text{,}{d}^{\prime })$ be a metric space. Prove that a family $\mathcal{S}\subseteq C(E\text{,}{E}^{\prime })$ , satisfying the uniform Lipschitz condition, that is

$\exists k>0\text{such that}\forall f\in \mathcal{S}\text{and}\forall p\text{,}q\in E\text{,}{d}^{\prime }(f(p)\text{,}f(q))\le \mathit{kd}(p\text{,}q)\text{,}$

is equicontinuous.

6.24

Let $(E\text{,}d)$ be a compact metric space and let $(E^{\prime }\text{,}{d}^{\prime })$ be a metric spaces. Prove that a family $\mathcal{S}\subseteq C(E\text{,}{E}^{\prime })$ , satisfying the uniform $\alpha$ -Hölder condition with the number $0<\alpha <1$ , that is

$\exists k>0\text{such that}\forall f\in \mathcal{S}\text{and}\forall p\text{,}q\in E\text{,}{d}^{\prime }(f(p)\text{,}f(q))\le \mathit{kd}{(p\text{,}q)}^{\alpha }\text{,}$

is equicontinuous.

6.25

Prove Theorem 6.29 for the space $C(E\text{,}{\mathbb{R}}^k)$ , where $k\in \mathbb{N}$ and $k\ge 2$ .

Hint: Consult with the proof of Lemma 4.6.

6.26

Show that $C(a\text{,}b)$ is a continuum set.

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C0-Semigroups and Application

In North-Holland Mathematics Studies, 2003

Example 2.1.2

Let X = C_ub (ℝ₊ ) be the space of all bounded and uniformly continuous functions from ℝ ₊ to ℝ, endowed with the sup-norm ||·||_∞, and let $\left\{S\left(t\right);t\ge 0\right\}\subseteq ℒ\left(X\right)$ be defined by

$\left[S\left(t\right)f\right]\left(s\right)=f\left(t+s\right)$

for each f ∈ X and each t, s ∈ ℝ₊. One may easily verify that {S(t); t ≥ 0} satisfies (i) and (ii) in Definition 2.1.1, and therefore it is a semigroup of linear operators. As in this specific case, the uniform continuity of the semigroup is equivalent to the equicontinuity of the unit ball in X, property which obviously is not satisfied, the semigroup is not uniformly continuous.

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Modern General Topology

In North-Holland Mathematical Library, 1985

Corollary

Let X be a compact uniform space. Then every continuous mapping of X into a uniform space Y is uniformly continuous.

Let(P) be a property of topological spaces or of uniform spaces. A uniform space X is called uniformly locally (P) if it has a uniform covering $\mathcal{U}$ each of whose elements has (P). In this connection, the following proposition may be of some interest.

H)

Let X be a connected, uniformly locally compact, uniform space. Then it is the union of countably many compact spaces, i.e., it is σ-compact.

Proof

Since X is uniformly locally compact, there is a uniform covering ${\mathcal{U}}_{\alpha }$ consisting of compact sets. Take a uniform covering ${\mathcal{U}}_{\beta }$ with ${\mathcal{U}}_{\beta }^{*}<{\mathcal{U}}_{\alpha }$ and a non-empty member U of ${\mathcal{U}}_{\beta }$ Then we put

and

(1) $\begin{array}{ll}{F}_{n+1}=\overline{S\left(F_n,{\mathcal{U}}_{\beta }\right),}\hfill & n=1,2,\mathrm{\dots .}\hfill \end{array}$

Then F ₁ is compact because

for some member $U^{\prime }$ of ${\mathcal{U}}_{\alpha }$ which is compact. We shall prove by induction on n that the F_n , n = 1, 2, …, are compact. Assume that F_n is compact. Then

for some $U_i\in {\mathcal{U}}_{\beta },$ i = 1, …, k. Since ${\mathcal{U}}_{\beta }^{*}<{\mathcal{U}}_{\alpha },$ we can choose $U_i^{\text{'}}\in {\mathcal{U}}_{\alpha },$ i = 1, …, k, for which

For these ${U^{\prime }}_i,$ we can prove that

Because, if $U\in {\mathcal{U}}_{\beta }$ and $U\cap F_n\ne 0,$ then $U\cap U_i\ne 0$ for some i, which implies that

$U\subset S\left(U_i,{\mathcal{U}}_{\beta }\right)\subset U_i^{\text{'}}\subset \cup \left\{U_i^{\text{'}}|i=1,\mathrm{\dots },k\right\}.$

Therefore

Since each ${U^{\prime }}_i$ is compact, it is closed and hence $\cup \left\{{U^{\prime }}_i|i=1,\dots ,k\right\}$ is a closed compact set. Therefore

$\begin{array}{ll}{F}_{n+1}\subset \cup \left\{{U^{\prime }}_i|i=1,\dots ,k\right\}\hfill & (\text{see}(1)).\hfill \end{array}$

This means that F _n+1 is compact since it is a closed subset of a compact set. Now, we can prove that ${\cup }_{n=1}^{\infty }{F}_n=X$ follows from the connectedness of X. For if $p\in {\cup }_{n=1}^{\infty }{F}_n,$ then p ∈ F_n for some n. Hence by (1)

Thus ${\cup }_{n=1}^{\infty }{F}_n$ is open. On the other hand, if p ≠ ∪ F_n , then

For, if we assume the contrary, then

This implies that p ∈ F_n (see (1)), which is a contradiction. Thus ${\cup }_{n=1}^{\infty }{F}_n$ is open and closed. Therefore it follows from the connectedness of X that

and hence X is the countable sum of the compact sets F_n , n = 1, 2, ….

A complete uniform space $\tilde{X}$ is called a completion of a uniform space X if X is unimorphic with a dense subspace of the uniform space $\tilde{X}.$ The position which is occupied by completion in the theory of uniform spaces is somewhat like that of compactification in the theory of topological spaces.

Now, let us consider the problem of constructing a completion of a given uniform space X. If X is a metric space, then by Example VI.6 it is isometrically imbedded in a complete metric space C ^*(X). Thus the closure $\overline{X}$ of X in C ^*(X) is a completion of X (see F)). If X is a general uniform space, then by use of the following proposition we can uniformly imbed X in the product of metric spaces. Therefore X is unimorphic to a subspace of a complete uniform space because the product space of complete uniform spaces is complete (see G)). Thus $\overline{X}$ in the product space is a completion of X.

I)

Every uniform space X is unimorphic to a subset of the product space of metric spaces.

Proof

Let us denote by $\left\{{{\rho }^{\prime }}_{\alpha }|\alpha \in A\right\}$ the collection of all uniformly continuous functions ${{\rho }^{\prime }}_{\alpha }$ over X × X which satisfy

$\begin{array}{ll}{{\rho }^{\prime }}_{\alpha }(p,q)={{\rho }^{\prime }}_{\alpha }(q,p)\ge 0,\hfill & {{\rho }^{\prime }}_{\alpha }(p,p)=0\hfill \end{array}$

and

${{\rho }^{\prime }}_{\alpha }(p,q)+{{\rho }^{\prime }}_{\alpha }(q,r)\ge {{\rho }^{\prime }}_{\alpha }(p,r).$

For each α ∈ A, the relation ${{\rho }^{\prime }}_{\alpha }(p,q)=0$ is an equivalence relation between two points p, q of X. Therefore classifying all points of X by use of this relation we obtain a decomposition ${\mathcal{D}}_{\alpha }$ of X. Define a function ${\rho }_{\alpha }(D,D^{\prime })$ on pairs of members of ${\mathcal{D}}_{\alpha }$ by

where p ∈ D, $p^{\prime }\in D^{\prime }.$ We can easily verify that ρ_α defined in this manner satisfies the conditions for a metric, and hence ${\mathcal{D}}_{\alpha }$ becomes a metric space which we denote by X _α. Moreover we denote by φ_α the natural mapping of X onto X_α, i.e.,

Now define a mapping φ of X into the product space P = Π {X_α | α ∈ A}

of the uniform spaces X _α as follows:

Since we can easily see that φ is a one-to-one uniformly continuous mapping, we shall prove only the uniform continuity of φ⁻¹.

Let $\mathcal{U}$ be a given uniform covering of X; then we choose a sequence $\left\{{\mathcal{U}}_n|n=1,2,\mathrm{\dots }\right\}$ of uniform coverings such that

For convenience of description, we put ${\mathcal{U}}_0=\left\{X\right\}$ . For p, q ∈ X, we define

$\begin{array}{l}\sigma \left(p,q\right)=\text{inf}\left\{2^{-n}|q\in S\left(p,{\mathcal{U}}_n\right)\right\},\hfill \\ {\rho }^{\prime }\left(p,q\right)=\text{inf}\left\{\sigma \left(p_0,p_1\right)+\sigma \left(p_1,p_2\right)+\mathrm{\dots }+\sigma \left(p_{k-1},p_k\right)|p_i\in X,i=1,\mathrm{\dots },k-1;p_0=p,p_k=q\right\}.\hfill \end{array}$

Then we can show that ${\rho }^{\prime }$ is a member of $\left\{{{\rho }^{\prime }}_{\alpha }|\alpha \in A\right\}$ . It is easy to prove $\begin{array}{llll}{\rho }^{\prime }\left(p,q\right)={\rho }^{\prime }\left(q,p\right)\ge 0,\hfill & {\rho }^{\prime }\left(p,p\right)=0\hfill & \text{and}\hfill & {\rho }^{\prime }\left(p,q\right)+{\rho }^{\prime }\left(q,r\right)\ge {\rho }^{\prime }\left(p,r\right)\hfill \end{array}$ from the definition of ${\rho }^{\prime }$ . To see its uniform continuity, assume

then $\sigma \left(p,p^{\prime }\right)\le 2^{-n},\sigma \left(q,q^{\prime }\right)\le 2^{-n}.$ . This implies

${\rho }^{\prime }\left(p^{\prime },q^{\prime }\right)\le \sigma \left(p^{\prime },p\right)+{\rho }^{\prime }\left(p,q\right)+\sigma \left(q,q^{\prime }\right)\le 2^{1-n}+{\rho }^{\prime }\left(p,q\right)$

and

i.e.,

proving that ${\rho }^{\prime }\left(p,q\right)$ is uniformly continuous. Thus we assume

Now, let us prove that

(2) $\sigma \left(p_0,p_1\right)+\sigma \left(p_1,p_2\right)+\dots +\sigma \left(p_{k-1},p_k\right)\ge \frac{1}{2}\sigma \left(p_0,p_k\right)$

for any choice of $p_0,p_1,\dots p_k,\in X.$

We use induction on the number k. For k = 1, the inequality (2) is obviously true. Assume it is true for all k < l. Put

and denote by m the largest number such that

$\sigma \left(p_0,p_1\right)+\dots +\sigma \left(p_{m-1},p_m\right)\le \frac{1}{2}s.$

Then

$\sigma \left(p_0,p_1\right)+\dots +\sigma \left(p_m,p_{m+1}\right)>\frac{1}{2}s,$

which implies

$\sigma \left(p_{m+1},p_{m+2}\right)+\dots +\sigma \left(p_{l-1},p_l\right)\le \frac{1}{2}s.$

Therefore it follows from the induction hypothesis that

$\begin{array}{ll}\frac{1}{2}\sigma \left(p_0,p_m\right)\le \frac{1}{2}s,\hfill & \frac{1}{2}\sigma \left(p_{m+1},p_l\right)\le \frac{1}{2}s.\hfill \end{array}$

On the other hand, $\sigma \left(p_m,p_{m+1}\right)\le s$ is obvious. Denote by n the smallest number such that 2⁻ⁿ ≤ s; then, in view of the definition of σ, we can assert

$\begin{array}{lll}\sigma \left(p_0,p_m\right)\le 2^{-n},\hfill & \sigma \left(p_m,p_{m+1}\right)\le 2^{-n},\hfill & \sigma \left(p_{m+1},p_l\right)\le 2^{-n}.\hfill \end{array}$

Because, e.g. if $\sigma \left(p_0,p_m\right)>2^{-n},$ then $\sigma \left(p_0,p_m\right)=2^{-t}$ and t < n. Thus 2^−t ≤ s, which contradicts the definition of n. Hence from the definition of σ it follows that

$\begin{array}{lll}{p}_m\in S\left(p_0,{\mathcal{U}}_n\right),\hfill & p_{m+1}\in S\left(p_m,{\mathcal{U}}_n\right),\hfill & p_1\in S\left(p_{m+1},{\mathcal{U}}_n\right).\hfill \end{array}$

Since ${\mathcal{U}}_n^{*}<{\mathcal{U}}_{n-1}$ , these imply

i.e.,

proving (2) in the case k = l. Thus (2) is generally true.

Suppose $q\notin S\left(p,\mathcal{U}\right);$ then by (2)

$\sigma \left(p_0,p_1\right)+\dots +\sigma \left(p_{k-1},p_k\right)\ge \frac{1}{2}\sigma \left(p_0,p_k\right)=\frac{1}{2}$

for p ₀ = p, p_k = q and for any choice of p ₁, …, p _k−1. Therefore, by the definition of $\begin{array}{ll}{\rho }^{\prime },\hfill & {\rho }^{\prime }\left(p,q\right)={{\rho }^{\prime }}_{\alpha }\left(p,q\right)\ge \frac{1}{2}\hfill \end{array}$ (see (1)). This means that if $\begin{array}{llll}{\phi }^{-1}\left(p^{\prime }\right)=p,\hfill & {\phi }^{-1}\left(q^{\prime }\right)=q\hfill & \text{and}\hfill & {\rho }_{\alpha }\left({p^{\prime }}_{\alpha },{q^{\prime }}_{\alpha }\right)<\frac{1}{2}\hfill \end{array}$ in X _α, then $q\in S\left(p,\mathcal{U}\right)$ in X. Thus φ⁻¹ is uniformly continuous, and accordingly, φ is unimorphic.

Now, we shall give a direct method of constructing a completion of a given uniform space X without imbedding X in the product space of metric spaces. Let X be a uniform space with a basis $\left\{{\mathcal{U}}_{\alpha }|\alpha \in A\right\}$ of uniformity. We call two Cauchy filters $ℱ\text{,}\mathcal{G}$ of X equivalent if for every α ∈ A, there is $U\in {\mathcal{U}}_{\alpha }$ satisfying $U\in ℱ\cap \mathcal{G}$ . We denote this equivalence relation by $ℱ\sim \mathcal{G}$ .

J)

If $ℱ\sim \mathcal{G}$ and $\mathcal{G}\sim \mathscr{H}$ for three Cauchy filters $ℱ,\mathcal{G}$ and $\mathscr{H}$ , then $ℱ\sim \mathscr{H}$ .

Proof

Given α ∈ A, we take β ∈ A such that ${\mathcal{U}}_{\beta }^{*}<{\mathcal{U}}_{\alpha }$ . Then there are $U\in {\mathcal{U}}_{\beta },U^{\prime }\in {\mathcal{U}}_{\beta }$ such that

Since

we get

and hence

This means that

i.e. $ℱ\sim \mathscr{H}.$

In view of J) we can classify all Cauchy filters of X and denote by $\tilde{X}$ the collection of all those classes. Suppose p is a point of X; then the filter $ℱ\left(p\right)=\left\{A|p\in A\right\}$ is clearly a Cauchy filter of X. We denote by ${\{ℱ\}}_p$ the class containing $ℱ\left(p\right)$ . Then there is a one-to-one mapping between the collection $\tilde{X}$ of those special classes and X. For brevity of description we regard X as a subset of $\tilde{X}$ identifying X with (i.e. ${\{ℱ\}}_p=p$ ). Now, let us assume that the given basis $\left\{{\mathcal{U}}_{\alpha }|\alpha \in A\right\}$ of the uniformity of X consists of open coverings of X. Generally for a given open set U of X, we denote by $\tilde{U}$ the subset of $\tilde{X}$ defined by

$\tilde{U}=\left\{q|q\in \tilde{X},U\in ℱ\text{ for every}ℱ\in q\right\}.$

put

Then it is easily seen that the restriction of ${\tilde{\mathcal{U}}}_{\alpha }$ to X is ${\mathcal{U}}_{\alpha }$ . Furthermore we can show:

K)

${\tilde{\mathcal{U}}}_{\alpha }$ is a covering of $\tilde{X}$ for each α ∈ A.

Proof

Let q be a given point of $\tilde{X}$ and choose β ∈ A for which ${\mathcal{U}}_{\beta }^{*}<{\mathcal{U}}_{\alpha }.$ Suppose that $ℱ$ is a member of q regarded as a class of Cauchy filters; then since $ℱ$ is Cauchy, F ⊂ U for some $F\in ℱ$ and $U\in {\mathcal{U}}_{\beta },$ i.e. $U\in ℱ$ . We take a member $U^{″}$ of ${\mathcal{U}}_{\alpha }$ for which

Let $\mathcal{G}$ be a given filter belonging to q; then it follows from $ℱ\sim \mathcal{G}$ that

Since $U,U^{\prime }\in ℱ,$ we obtain

and hence

Since $U^{\prime }\in \mathcal{G},U^{″}\in \mathcal{G}.$

Thus we have proved that there is a $U^{″}\in {\mathcal{U}}_{\alpha }$ such that $U^{″}\in \mathcal{G}$ for every $\mathcal{G}\in q.$ . Therefore $q\in \widetilde{U^{″}}$ follows from the definition of $\widetilde{U^{″}}$ . Hence ${\tilde{\mathcal{U}}}_{\alpha }$ covers $\tilde{X}$ .

L)

if ${\mathcal{U}}_{\alpha }^{*}<{\mathcal{U}}_{\beta }$ in X then ${\tilde{\mathcal{U}}}_{\alpha }^{*}<{\tilde{\mathcal{U}}}_{\beta }$ in $\tilde{X}$ .

Proof

We note first that if U ⊂ V holds for open sets U, V of X, then $\tilde{U}\subset \tilde{V}$ . Now, suppose that $U,U^{\prime }\in {\mathcal{U}}_{\alpha }$ and $U\cap U^{\prime }=0.$ Then each filter cannot contain U and $U^{\prime }$ at the same time. This implies that $\tilde{U}\cap \widetilde{U^{\prime }}=0.$ Therefore, for a given member ${\tilde{U}}_0$ of ${\tilde{\mathcal{U}}}_{\alpha }$ we obtain

$S\left({\tilde{U}}_0,{\tilde{\mathcal{U}}}_{\alpha }\right)=\cup \left\{\tilde{U}|U\cap U_0\ne 0,U\in {\mathcal{U}}_{\alpha }\right\}\subset {\tilde{U}}^{\prime }$

for some $U^{\prime }\in {\mathcal{U}}_{\beta }$ satisfying $S\left(U_0,{\mathcal{U}}_{\alpha }\right)\subset U^{\prime }$ since ${\mathcal{U}}_{\alpha }^{*}<{\mathcal{U}}_{\beta }.$ hence ${\tilde{\mathcal{U}}}_{\alpha }^{*}<{\tilde{\mathcal{U}}}_{\beta }.$

M)

Suppose q, $q^{\prime }$ are two different points of $\tilde{X}$ . Then $q^{\prime }\notin S\left(q,{\tilde{\mathcal{U}}}_{\alpha }\right)$ for some α ∈ A.

Proof

Let $ℱ\in q$ and ${ℱ}^{\prime }\in q^{\prime }$ . Then, since $ℱ\sim ℱ\prime$ , there is a ${\mathcal{U}}_{\alpha }$ each member of which does not belong to $ℱ\cap {ℱ}^{\prime }$ . Let U be a given member of ${\mathcal{U}}_{\alpha }$ such that $q\in \tilde{U}$ . Then by the definition of $\tilde{U}$ , $U\in ℱ$ and hence $U\notin {ℱ}^{\prime },$ i.e. $q^{\prime }\notin \tilde{U}$ . Therefore $q^{\prime }\notin S\left(q,{\tilde{\mathcal{U}}}_{\alpha }\right).$

It follows from K) and M) that $\left\{{\tilde{\mathcal{U}}}_{\alpha }|\alpha \in A\right\}$ is a basis of uniformity of $\tilde{X}$ . Since the thus defined uniform space $\tilde{X}$ clearly contains X as a subspace, we shall show that $\tilde{X}$ is a complete uniform space satisfying $\overline{X}=\tilde{X}$ . For that purpose we need the following proposition which is valid for any uniform space.

N)

Let p be a cluster point of a Cauchy filter basis $ℱ$ . Then $ℱ\to p.$

Proof

Given α ∈ A, we choose β ∈ A for which ${\mathcal{U}}_{\beta }^{*}<{\mathcal{U}}_{\alpha }.$ Since $ℱ$ is Cauchy, F ⊂ U for some $F\in ℱ$ and $U\in {\mathcal{U}}_{\beta }$ . On the other hand, since $p\in \overline{F}$ , there is $U^{\prime }\in {\mathcal{U}}_{\beta }$ with $p\in U^{\prime }$ and $U^{\prime }\cap U\ne 0$ . Hence

$\begin{array}{ll}\left\{p\right\}\cup F\subset U^{\prime }\cup U\subset U^{″}\hfill & \text{for some}{U}^{″}\in {\mathcal{U}}_{\alpha }.\hfill \end{array}$

Thus $F\subset S\left(p,{\mathcal{U}}_{\alpha }\right).$ Since $\left\{S\left(p,{\mathcal{U}}_{\alpha }\right)|\alpha \in A\right\}$ is a nbd basis of p, we can conclude that $ℱ\to p.$

Proof

Given $q_0\in \tilde{X}$ and α ∈ A. Suppose that

Then

This implies that $q_0\in \overline{X}$ , proving the assertion.

Proof

Let $\mathcal{G}$ be a given Cauchy filter of $\tilde{X}$ . Then $ℱ=\{G\cap X|G\in \mathcal{G},$ and G is open in $\tilde{X}$ } is a Cauchy filter basis of X. We denote by q ₀ the class which contains the Cauchy filter ${ℱ}^{\prime }$ of X generated by $ℱ$ . We regard q ₀ as a point of $\tilde{X}$ , to prove that $\mathcal{G}\to q_0.$ Given ${\tilde{\mathcal{U}}}_{\alpha },$ we suppose that $\tilde{U}$ is a member of ${\tilde{\mathcal{U}}}_{\alpha }$ such that $q_0\in \tilde{U}$ . Then $U\in {ℱ}^{\prime },$ which implies that $U\cap F\ne 0$ for every $F\in ℱ.$ Therefore $\tilde{U}\cap G\ne 0$ for every open $G\in \mathcal{G},$ which means that q ₀ is a cluster point of the Cauchy filter basis ${\mathcal{G}}^{\prime }=\{G|G\in \mathcal{G},$ and G is open in $\tilde{X}$ }. It follows from N) that ${\mathcal{G}}^{\prime }\to q_0$ Therefore $\mathcal{G}\to q_0$ . This proves that $\tilde{X}$ is complete.

Now pursuing the analogy between completion and compactification, we can assert that every real-valued uniformly continuous function over X can be extended to a uniformly continuous function over $\tilde{X}$ . In fact we can prove more generally the following proposition.

Q)

Let $X^{\prime }$ be a dense subspace of a uniform space X, i.e. $\overline{X^{\prime }}=X$ . Then every uniformly continuous mapping f of $X^{\prime }$ into a complete uniform space Y can be extended to a uniformly continuous mapping of X into Y ¹ .

Proof

Let p be a given point of X. Then

$ℱ=\left\{U\left(p\right)\cap X^{\prime }|U\left(p\right)\text{ is a nbd of}p\text{ in}X\right\}$

is a Cauchy filter of $X^{\prime }$ . We can assert that

$\mathcal{G}=\left\{G|G\supset f\left(F\right)\text{ for some}F\in ℱ\right\}$

is a Cauchy filter of Y. To see it, suppose that ${\mathcal{V}}_{\alpha }$ is a given uniform covering of Y; then by the uniform continuity of f,

is a uniform covering of X. Therefore $F\subset f^{-1}\left(V\right)$ for some $F\in ℱ$ and $V\in {\mathcal{V}}_{\alpha }$ . This implies that $f\left(F\right)\subset V$ , i.e. $\mathcal{G}$ is a Gauchy filter of Y. Since Y is complete, $\mathcal{G}$ converges to a point q ∈ Y. Put g(p) = q. The mapping g defined in this manner is clearly an extension of f over X. The only thing to be proved is that g is uniformly continuous over X. Suppose that ${\mathcal{V}}_{\alpha }$ is a given uniform covering of Y. Then take an open uniform covering ${\mathcal{V}}_{\beta }$ with ${\mathcal{V}}_{\beta }^{*}<{\mathcal{V}}_{\alpha }.$

Now, we can prove that

(1) $\left\{X-\overline{X^{\prime }-f^{-1}\left(V\right)}|V\in {\mathcal{V}}_{\beta }\right\}<g^{-1}\left({\mathcal{V}}_{\alpha }\right)\text{ in}X.$

For, If $p\in X-\overline{X^{\prime }-f^{-1}\left(V\right)}\text{ for}V\in {\mathcal{V}}_{\beta },$ then

is a nbd of p in X satisfying

since f ⁻¹(V) is open in $X^{\prime }$ . Hence

for a member $V^{\prime }$ of ${\mathcal{V}}_{\alpha }$ such that $S\left(V,{\mathcal{V}}_{\beta }\right)\subset V^{\prime }.$ Since p is a given point of $X-\overline{X^{\prime }-f^{-1}\left(V\right)}$ , we get $g\left(X-\overline{X^{\prime }-f^{-1}\left(V\right)}\right)\subset V^{\prime }$ , i.e.,

proving (1). Since f is uniformly continuous, $f^{-1}\left({\mathcal{V}}_{\beta }\right)$ is a uniform covering of $X^{\prime }$ . This means that there is an open uniform covering ${\mathcal{U}}_{\beta }$ of X such that the restriction ${{\mathcal{U}}^{\prime }}_{\beta }$ of ${\mathcal{U}}_{\beta },$ to $X^{\prime }$ is a refinement of $f^{-1}\left({\mathcal{V}}_{\beta }\right).$ In other words, for a given member U of ${\mathcal{U}}_{\beta },$ there is a member V of ${\mathcal{V}}_{\beta }$ for which

Therefore

(note that U is open), i.e.,

proving

This relation combined with (1) implies that $g^{-1}\left({\mathcal{V}}_{\alpha }\right)$ is a uniform covering of X, proving that g is a uniformly continuous mapping.

To end our discussion on completion, let us verify the uniqueness of completion as follows:

R)

Let $X^{\prime }$ and $\stackrel{\_}{X}$ be completions of a uniform space X. Then $X^{\prime }$ and $\stackrel{\_}{X}$ are unimorphic by a mapping which keeps X fixed.

Proof

We consider the identity mapping f of X onto X. Then by Q) we can extend f to a uniformly continuous mapping g of $X^{\prime }$ into $\stackrel{\_}{X}$ . In the same way, we can extend f ⁻¹ to a uniformly continuous mapping h of $\stackrel{\_}{X}$ into $X^{\prime }$ . Suppose that $p\in \widetilde{X,}\text{undefined}q\in \stackrel{\_}{X}$ and g(p) = q. Let

$\mathcal{G}=\left\{V\left(q\right)\cap X|V\left(q\right)\text{ is a nbd of}q\text{ in}\widehat{X}\right\}.$

Putting

$ℱ=\left\{F|F\subset \tilde{X},F\supset f^{-1}\left(G\right)\text{ for some}G\in \mathcal{G}\right\},$

we get a Cauchy filter $ℱ$ of $X^{\prime }$ . As seen in the proof of Q),

On the other hand, we can assert that $ℱ\to p$ in $X^{\prime }$ . To see it, let U(p) be a given nbd of p in $X^{\prime }$ . Then by the definition of g,

Hence it follows from the definition of $\mathcal{G}$ that

$\begin{array}{ll}G\cap f\left(U\left(p\right)\cap X\right)\ne 0\hfill & \text{for every}G\in \mathcal{G},\hfill \end{array}$

which implies

$\begin{array}{ll}{f}^{-1}\left(G\right)\cap U\left(p\right)\ne 0\hfill & \text{for every}G\in \mathcal{G}\text{ in}\tilde{X}.\hfill \end{array}$

Thus $p\in \overline{f^{-1}\left(G\right)}$ , which means that p is a cluster point of $ℱ$ . Since $ℱ$ is a Cauchy filter, by N), $ℱ\to p.$ Thus from (1) we can conclude that h(q) = p, i.e. h = g ⁻¹. This proves that g is a unimorphic mapping of $X^{\prime }$ onto $\stackrel{\_}{X}$ , proving the proposition.

Combining O), P), Q) and R) we obtain the following theorem.

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Handbook of the Geometry of Banach Spaces

Robert Deville , Nassif Ghoussoub , in Handbook of the Geometry of Banach Spaces, 2001

4.2 The sub-differential of the sum

Formulae for the subdifferential of the sum of two functions defined on a Banach space have been investigated by various authors (Ioffe [55], Fabian [34,35], Deville and El Haddad [27]). We present here an extension, observed by Bachir [5], of a result obtained by Borwein and Zhu [9] in the context of Banach spaces admitting an equivalent Fréchet-differentiable norm. We need the following:

Definition 4.1

1. Let u ₁, $u_2:X\to \text{R}\cup \left\{+\infty \right\}$ be lower semicontinuous functions. We say that the pair (u ₁, u ₂) is locally uniformly lower semicontinuous if, for every x ∈ X and every uniformly continuous function φ : X × X → R, there exists r > 0 such that,

$\begin{array}{l}\underset{y\in B}{\inf }{u}_1\left(y\right)+u_2\left(y\right)+\varphi \left(y,y\right)\\ =\underset{\eta \to 0}{\lim }\inf \left\{u_1\left(y\right)+u_2\left(z\right)+\varphi \left(y,\ge \right);||y-z||<\eta ,y,z\in B\right\}.\end{array}$

where B = B_X (x, r).

This condition is clearly stable if we perturb u ₁ and u ₂ by uniformly continuous functions, and is always satisfied when dim(X) < +∞, or when one of the functions is locally uniformly continuous.

Theorem 4.2

Let X be a Banach space satisfying (H1). Let $u_1,u_2:X\to \text{R}\cup \left\{+\infty \right\}$ be lower semicontinuous functions. Assume that the pair (u ₁, u ₂) is locally uniformly lower semicontinuous. Suppose that x ₀ ∈ X and p ∈ D ⁻(u ₁ + u ₂)(x ₀) ore given. Then, for each ε > 0, there exist $x_1,x_2\in X,p_1\in D^{-}{u}_1\left(x_1\right)$ and $p_2\in D^{-}{u}_2\left(x_2\right)$ such that:

(1)

$||x_1-x_0||<\epsilon ,||x_2-x_0||<\epsilon and||x_1-x_2||.\left(||p_1||+||p_2||\right)<\epsilon ,$

(2)

$\left|u_1\left(x_1\right)-u_1\left(x_0\right)\right|<\epsilon and\left|u_2\left(x_2\right)-u_2\left(x_0\right)\right|<\epsilon$ , and

(3)

$||p_1+p_2-p||<\epsilon$ .

Theorem 4.3 is a non trivial consequence of Theorem 4.1 on the minimization of the sum of two functions. A Rademacher–Preiss type theorem for uniformly continuous functions in spaces which admit a smooth bump function can be deduced from this formula. Let us recall that according to Rademacher theorem, every Lipschitz continuous function in R ⁿ is differentiable almost everywhere. Preiss [62] has extended this result to an infinite dimensional setting. He proved in particular that if X is an Asplund space (in particular if X satisfies (H1)), then every locally Lipschitz continuous real valued function defined on X is differentiable on a dense subset of X. The following result from [27] can be seen as a weak form of Preiss theorem for uniformly continuous functions.

Corollary 4.3

Let X be a Banach space satisfying (H1). Let u be a uniformly continuous function defined on X. Then for every x ∈ X and every ε > 0, there exist x ₁, x ₂ ∈ X, there exist $p^{-}\in D^{-}u\left(x_1\right)and{p}^{+}\in D^{+}u\left(x_2\right)$ such that:

(1)

$||x_1-x||<\epsilon and||x_2-x||<\epsilon ,$

(2)

$\left|u\left(x_1-u\left(x\right)\right)\right|<\epsilon and\left|u\left(x_2-u\left(x\right)\right)\right|<\epsilon ,$

(3)

$||p^{-}-p^{+}||<\epsilon .$

In order to prove Corollary 4.3, it is enough to apply Theorem 4.2 with u ₁ = u and u ₂ = –u, and to observe that $D^{-}\left(-u\right)\left(x_2\right)=D^{+}u\left(x_2\right)$ . Let us here stress the fact that Preiss' result is considerably harder to prove.

Problem 4

If the function u of Corollary 4.3 is nowhere differentiable, then the points x ₁ and x ₂ are necessarily different. It is unknown whether one can take p ⁻ = p ⁺ when u is an arbitrary uniformly continuous function (the answer to this question is positive if dim X = 1).

Problem 5

Counterexamples show that the assumption "(u ₁, u ₂) is locally uniformly lower semicontinuous" cannot be dropped in Theorem 4.1 (see for instance [29]). These counterexamples involve extended valued functions. It is an open problem to know whether a formula for the subdifferential of the sum of two lower semi continuous, real valued functions in infinite dimensions always holds true.

Remark 4.2. Theorem 4.2 will be applied in the next section in the proof of uniqueness of the viscosity solution of a particular Hamilton–Jacobi equation. The conclusion " $||x_1-x_2||.\left(||p_1||+||p_2||\right)<\epsilon$ " in Theorem 4.2 is usually needed in the case of Hamilton–Jacobi equations coming from optimal control or differential games.

Most results can be established for weaker types of differentiability. Indeed, let u : X → R ∪ {+∞} be an arbitrary function. If x ∈ X, we may define for example the G-viscosity subdifferential of u at x:

$D_G^{-}u\left(x\right)=\left\{g^{\prime }\left(x\right);g:X\to \text{R}\text{is\hspace{1em}G}\widehat{\text{a}}\text{teaux-diff}\text{.}\text{and}u-g\text{has\hspace{1em}a\hspace{1em}local\hspace{1em}min}\text{.}\text{at}x\right\}.$

The G-viscosity superdifferential of u at x is defined similarly. The following result is the analogue of Theorem 4.2.

Proposition 4.3

Let X be a Banach space on which there exists a Lipschitz continuous, Gâteaux-differentiable bump function b on X such that b′ is norm to weak^* continuous. Let u ¹, u ₂ be two real valued functions on X such that u ₁ is lower semi continuous and u ₂ is uniformly continuous. Suppose that x ₀ ∈ X and $p\in D_G^{-}\left(u_1+u_2\right)\left(x_0\right)$ are given. Then, for each ε > 0 and each weak^* neighbourhood V of p, there exist $x_1,x_2\in X,p_1\in D_G^{-}{u}_1\left(x_1\right)and{p}_2\in D_G^{-}{u}_2\left(x_2\right)$ such that:

(1)

$||x_1-x_0||<\epsilon and||x_2-x_0||<\epsilon ,$

(2)

$\left|u_1\left(x_1\right)-u_1\left(x_0\right)\right|<\epsilon and\left|u_2\left(x_2\right)-u_2\left(x_0\right)\right|<\epsilon ,$

(3)

$p_1+p_2\in V.$

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Signal and Image Representation in Combined Spaces

Charles K. Chui , Chun Li , in Wavelet Analysis and Its Applications, 1998

1 Introduction

It is well known that the coefficient sequence of the Fourier series expansion of a periodic signal (or function) is often used to characterize the order of smoothness of the signal itself. This so−called Littlewood−Paley approach to wavelet series expansions (often called discrete wavelet transforms, DWT) is also well documented in the wavelet literature (see, for instance, the monograph [10] of Y. Meyer). This study, however, is within the L^p theory of wavelet analysis. Based on the pioneer work of Donoho [5], we recently developed in [2] a parallel theory of the DWT for the space C_u of bounded and uniformly continuous functions on R. The basic functions that generate the local series expansions are interpolatory wavelets, such as those of Micchelli (see [2]) and Donoho [5]; and the coefficients are functional wavelet transforms, FnWT, of the functions in C_u under investigation. The importance of the FnWT is that it reveals the local details, just as the DWT does, of the functions under investigation by using their discrete function values directly, by means of Lagrange interpolation.

This paper may be considered a continuation of our work developed in [2]. The objective here is to characterize the smoothness of functions (or signals) in C_u ∩ L^p (1 ≤ p ≤ ∞) in terms of the wavelet coefficients of the interpolatory wavelet series representation that are used in the process of interpolatory wavelet decompositions. This characterization will involve Besov spaces as well as Sobolev spaces. We emphasize that the important role of the duals of the interpolatory wavelets is mainly due to the property of vanishing moments to be described in Section 2. In this respect, our point of view and mathematical analysis are quite different from those in Donoho [5].

This paper is organized as follows. In Section 2 we will discuss the notion of interpolatory scaling functions, wavelets, and their duals as introduced in [2], and will describe some of their basic properties that are needed for stating and establishing our main results. Unlike the L ² situation, the duals here are not functions in the same space as the wavelets and the original functions to be analyzed. In fact, they are distributions generated by the Dirac functional. The main results of this paper will be established in Sections 3 and 4.

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Sobolev Spaces*

Aleksander Pełczyński , Michał Wojciechowski , in Handbook of the Geometry of Banach Spaces, 2003

1 Classical Sobolev spaces

Let ${\partial }^{\alpha }=\frac{{\partial }^{\left|\alpha \right|}}{\partial x_1{}^{{\alpha }_1}\partial x_2{}^{{\alpha }_2}\dots \partial x_n{}^{\alpha n}}$ denote the partial derivative corresponding to the multi-index $\alpha ={\left({\alpha }_j\right)}_{j=1}^n\in {\mathbb{Z}}_{+}^n$ where ${\mathbb{Z}}_{+}:=\mathbb{N}\cup \left\{0\right\}$ ; here $\left|\alpha \right|={\displaystyle {\sum }_{j=1}^n{\alpha }_j}$ is the order of the derivative ∂ ^α ; we use the convention $x^{\alpha }:={\displaystyle {\prod }_{j=1}^n{x}_j^{{\alpha }_j}}$ . Define α ⪯ β ≡_def α_j ≤ β_j for j = 1, 2,…, n. If Ω is an open set in ℝ ⁿ then $\overline{\Omega }$ denotes the closure of Ω and bd $\Omega =\overline{\Omega }\backslash \Omega$ denotes the boundary of Ω. The set of scalars is denoted by $\mathbb{K}$ , it is, either ℝ – the real numbers, or ℂ – the complex numbers. Let $\mathcal{D}\left(\Omega \right)$ be the space of all infinitely many times differentiable scalar-valued functions $ƒ:\Omega \to \mathbb{K}$ with compact support, supp $ƒ=\overline{\left\{x:ƒ\left(x\right)\ne 0\right\}}\subset \Omega$ . A function $g:\Omega \to \mathbb{K}$ is said to be the α-th distributional partial derivative of an $ƒ:\Omega \to \mathbb{K}$ , in symbol g = D^α ƒ, provided

${{\displaystyle {\int }_{\Omega }g\varphi \text{d}x=\left(-1\right)}}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }f{\partial }^{\alpha }\varphi }\text{d}x\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for}\varphi \in \mathcal{D}\left(\Omega \right).$

Here and in the sequel ∫… dx denotes the integration against the n-dimensional Lebesgue measure λ_n ; by L^p (Ω) we denote L^p (Ω, λ_n ).

If a partial derivative of ƒ is continuous on Ω then the corresponding distributional derivative of ƒ coincides with the partial derivative. We admit that the (distributional) derivative of order 0 of a function ƒ coincides with ƒ.

Let 1 ≤ p ≤ ∞ and k = 0, 1, …. Let us put

$L_{\left(k\right)}^p\left(\Omega \right)=\left\{ƒ:\Omega \to \mathbb{K}:D^{\alpha }ƒ\text{\hspace{0.17em}exists\hspace{0.17em}and}{D}^{\alpha }ƒ\in L^p\left(\Omega \right)\text{\hspace{0.17em}for}\left|\alpha \right|\le k\right\}.$

We equip $L_{\left(k\right)}^p\left(\Omega \right)$ with the norm

${\Vert ƒ\Vert }_{\Omega ,\left(k\right),p}=\{\begin{array}{ll}{\left({\displaystyle \sum {}_{\left|\alpha \right|\le k}{\displaystyle {\int }_{\Omega }{\left|D^{\alpha }\text{ƒ}\left(x\right)\right|}^p\text{d}x}}\right)}^{1/p},\hfill & \text{for}1\le p<\infty \hfill \\ {\max }_{\left|\alpha \right|\le k}{\text{essup}}_{x\in \Omega }\left|D^{\alpha }\text{ƒ}\left(x\right)\right|,\hfill & \text{for}p=\infty \hfill \end{array}$

By $C_0^{\left(k\right)}\left(\Omega \right)$ we denote the closure of $\mathcal{D}\left(\Omega \right)$ in the norm ∥·∥ _Ω,(k),∞, and by C ^(k)(Ω) a subspace of $L_{\left(k\right)}^{\infty }\left(\Omega \right)$ consisting of functions which together with their partial derivatives of orders ≤ k are uniformly continuous and vanishes at infinity (for unbounded Ω). Clearly

$C_0^{\left(k\right)}\left(\Omega \right)\subseteq C^{\left(k\right)}\left(\Omega \right)\subset L_{\left(k\right)}^{\infty }\left(\Omega \right).$

Warning

Usually spaces in sup norms are defined on closed subsets of ℝ ⁿ . However the Sobolev spaces in L^p -norms are naturally defined on open subsets of ℝ ⁿ . To unify domains of functions we work with, we have defined C ^(k) (Ω ) as spaces of uniformly continuous functions on open Ω; the functions uniquely extend on the closure of Ω. On the other hand we work primarily with separable spaces thus we add the condition of vanishing at infinity. This condition is meaningful only for unbounded domains. Thus in our notation $C^{\left(k\right)}\left({ℝ}^n\right)=C_0^{\left(k\right)}\left({ℝ}^n\right)$ and $C^{\left(k\right)}\left(I^n\right)=C^{\left(k\right)}\left(\overline{I^n}\right)$ . Here and in the sequel Iⁿ = (1/2, 1/2) ⁿ ⊂ ℝ ⁿ .

The spaces $L_{\left(k\right)}^p\left(\Omega \right)\left(1\le p\le \infty \right)$ , $C_0^{\left(k\right)}\left(\Omega \right)$ , C ^(k) (Ω) for k = 1, 2,… are called classical Sobolev spaces. A routine argument gives:

Proposition 1

(i)

The classical Sobolev spaces are Banach spaces;

(ii)

$C_0^{(k)}(\Omega ),C^{\left(K\right)}\left(\Omega \right)$ and $L_{\left(k\right)}^p$ for 1 ≤ p < ∞ are separable.

For regular domains Ω ⊂ ℝ ⁿ Sobolev spaces in L^p norms (1 ≤ p < ∞) can be defined as completion of C ^∞-functions in the corresponding norms. Precisely we have

Proposition 2

$C^{\infty }\left(\Omega \right):={\cap }_{k=1}^{\infty }{C}^{\left(k\right)}\left(\Omega \right)$ is dense in $L{}_{\left(k\right)}^p\left(\Omega \right)$ in the following cases

(i)

Ω = ℝ ⁿ (cf. [88], Chapter V, §2, Proposition 1);

(ii)

there is a linear extension operator from $L{}_{\left(k\right)}^p\left(\Omega \right)$ to $L{}_{\left(k\right)}^p\left({ℝ}^n\right)$ which takes C ^∞ (Ω) into C ^∞ (ℝ ⁿ );

(iii)

Ω is bounded and has segment property, i.e., every x ∈ bd Ω has an open neighbourhood U_x in ℝ ⁿ and there exists a non-zero vector y_x such that for every $z\in \overline{\Omega }\cap U_x$ one has z + t y_x ∈ Ω for 0 < t < 1 (cf. [1], Chapter III, Theorem 3.18).

Remarks

(1)

In Proposition 2 one can replace C ^∞ (Ω) by its subspace consisting with functions whose unique continuous extension to $\overline{\Omega }$ has compact support.

(2)

A theorem of Stein (cf. [88], Chapter VI, §3, Theorem 5) provides for a large class of domains in ℝ ⁿ a construction of linear extension operators from $L{}_{\left(k\right)}^p\left(\Omega \right)$ to $L{}_{\left(k\right)}^p\left({ℝ}^n\right)$ taking C ^∞ (Ω) into C ^∞ (ℝ ⁿ ).

(3)

Proposition 2 extends also on bounded domains with so called cone property (cf. [62], § 1.1.9 for definition, and § 1.1.6 and § 1.1.9 for the proof).

(4)

Proposition 2 extends to Sobolev spaces on compact manifolds.

We outline briefly how to define Sobolev spaces on manifolds. For simplicity let M be an n-dimensional compact Euclidean C^k -manifold. Let ${\left({\varphi }_j,{\mathcal{O}}_j\right)}_{j\in A}$ be a finite atlas of M consisting of open sets ${\mathcal{O}}_j$ and homeomorphisms ${\varphi }_j:{\mathcal{O}}_j\underrightarrow{\text{into}}{ℝ}^n$ such that ${\Omega }_j:={\varphi }_j({\mathcal{O}}_j)$ are open subsets of ℝ ⁿ whose closures are compact. Assume furthermore that

${\varphi }_j{\varphi }_i^{-1}\in C^{\left(k\right)}\left({\Omega }_j\cap {\Omega }_i\right)\left(\left(j,i\right)\in A\times A\right).$

For 1 ≤ p ≤ ∞ we put

$\begin{array}{l}{L}_{\left(k\right)}^p\left({\rm M}\right)=\left\{ƒ:{\rm M}\to \mathbb{K}\text{:}ƒ\varphi {}_j^{-1}\in L_{\left(k\right)}^p\left({\Omega }_j\right)\text{for}j\in A\right\},\\ {\Vert ƒ\Vert }_{{\left({\varphi }_j,{\mathcal{O}}_j\right)}_{j\in A},\left(k\right),p}=\{\begin{array}{ll}{\left({\displaystyle \sum {}_{j\in A}{\Vert ƒ{\varphi }_j^{-1}\Vert }_{{\Omega }_j,\left(k\right),p}^p}\right)}^{1/p},\hfill & \text{for}1\le p<\infty ,\hfill \\ {\max }_{j\in A}{\Vert ƒ\Vert }_{{\Omega }_j,\left(k\right),\infty ,}\hfill & \text{for}p=\infty .\hfill \end{array}\end{array}$

The definition of C ^(k) (M) is analogous.

Note that the topology of $L{}_{\left(k\right)}^p\left(M\right)$ does not depend of a particular choice of an atlas; for different atlases we get equivalent norms. In the case where M is a Lie group or a homogeneous subspace of a Lie group (we assume that the group operation is compatible with the differentiable structure of M, i.e., translation by element of the group are diffeomorphisms) then the norm of $L{}_{\left(k\right)}^p$ can be naturally defined in terms of the Haar measure of the group and partial derivatives defined by elements of the Lie algebra of the group. This remark also applied to unimodular locally compact Lie groups.

The most useful special models of Sobolev spaces on manifolds are the spaces on the groups ℝ ⁿ and the tori ${\mathbb{T}}^n$ , and the spaces on the Euclidean spheres ${\mathbb{S}}_n$ which are homogeneous spaces of the orthogonal groups. Since the spaces under consideration are translation invariant with respect to the group action, we can use powerful tools of Harmonic Analysis to study their structure. The torus ${\mathbb{T}}^n$ is usually identified with spaces ${ℝ}^n$ of 1-periodic functions with 1-periodic derivatives with respect to each variable.

We end this section by introducing Sobolev spaces of measures – B V _(k) (Ω). For a measure μ denote by υ(μ) the positive measure being the total variation of μ (cf. [25], Vol. I, Chapter III, §1, Definition 1). M(Ω) stands for the Banach space of all scalar-valued Borel measures μ on Ω with bounded total variation with the norm ∥μ∥ _{M (Ω)} = υ(μ)(Ω). A measure ν is said to be the distributional derivative of a measure μ corresponding to the multi-index α, in symbols D^α μ = ν provided

$\begin{array}{cc}{\displaystyle {\int }_{\Omega }{\partial }^{\alpha }\phi \text{d}\mu ={\left(-1\right)}^{\alpha }}{\displaystyle {\int }_{\Omega }\phi \text{d}}\nu & \text{for}\phi \in \mathcal{D}\left(\Omega \right)\end{array}.$

We admit D ⁰ (μ) = μ.

For an open Ω ⊂ ℝ ⁿ and for k = 1, 2, … by B V _(k) (Ω) we denote the space of all μ ∈ B V (Ω) such that D^α (μ) exists and belongs to M (Ω) for 0 ≤ |α| ≤ k, equipped with the norm

${\Vert \mu \Vert }_{B{V}_{\left(k\right)}\left(\Omega \right)}={\displaystyle \sum _{0\le \left|\alpha \right|\le k}{\Vert D^{\alpha }\Vert }_{M\left(\Omega \right)}.}$

The elements of B V _(k) (Ω) can be regarded as functions on Ω. Precisely, it is not hard to show (cf., e.g., [81]) that if μ ∈ B V_k (Ω) then μ and all the distributional derivatives D^α (μ) for |α| < k are absolutely continuous with respect to the Lebesgue measure λ_n ; thus it is natural to identify them with functions in L ¹ (Ω). The space $L{}_{\left(k\right)}^1\left(\Omega \right)$ can be identified with the subspace of B V _(k) (Ω) being the image of the isometric embedding ƒ → ƒ · λ_n of $L{}_{\left(k\right)}^1\left(\Omega \right)$ into B V _(k) (Ω). The space B V ₍₁₎ (Ω) is often called the space of functions of bounded variation on Ω. For further information on spaces of functions with bounded variation the reader is referred to the books [27,62,104].

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Handbook of Dynamical Systems

Björn Sandstede , in Handbook of Dynamical Systems, 2002

Example 1 (Reaction-diffusion systems)

Let D be a diagonal N × N matrix with positive entries and $F:R^N\to R^N$ be a smooth function. Consider the reaction-diffusion equation

(2.9) $\begin{array}{ll}{U}_t=D{U}_{xx}+F(U),\hfill & x\in R,U\in R^N,\hfill \end{array}$

posed on the space $\chi =C_{unif}^0\left(R,R^N\right)$ of bounded, uniformly continuous functions. In the moving frame ξ = x - ct, the system (2.9) is given by

(2.10) $\begin{array}{ll}{U}_t=D{U}_{\xi \xi }+c{U}_{\xi }+F(U),\hfill & \xi \in R,U\in R^N.\hfill \end{array}$

Suppose that U(ξ, t) = Q(ξ) is a stationary solution of (2.10) such that

(2.11) $\begin{array}{ll}D{Q}_{\xi \xi }(\xi )+c{Q}_{\xi }(\xi )+F\left(Q(\xi )\right)=0,\hfill & \xi \in R.\hfill \end{array}$

The eigenvalue problem associated with the linearization of (2.10) about Q(ξ) is given by

(2.12) $\lambda U=D{U}_{\xi \xi }+c{U}_{\xi }+{\partial }_{U}F(Q)U=:LU.$

This eigenvalue problem can be cast as

$\begin{array}{lll}\left(\begin{array}{l}{U}_{\xi }\hfill \\ V_{\xi }\hfill \end{array}\right)& =& \left(D^{-1}(\lambda U-{\partial }_U\stackrel{V}{F}(Q(\xi ))U-cV)\right)\\ & =& \left(D^{-1}(\lambda -\partial \stackrel{0}{U}F(Q(\xi )))-c\stackrel{\text{id}}{D^{-1}}\right)\left(\begin{array}{l}U\hfill \\ V\hfill \end{array}\right)\end{array}$

which we write as

(2.13) $\begin{array}{ll}{u}_{\xi }=A(\xi ;\lambda )u=\left(\stackrel{\sim }{A}(\xi )+\lambda B\right)u,\hfill & u\in R^n=R^{2N}\hfill \end{array},$

with u = (U, V) and

$\begin{array}{ll}\stackrel{\sim }{A}(\xi )=\left(-D^{-1}\partial \stackrel{0}{}F(Q(\xi ))-c\stackrel{\text{id}}{D^{-1}}\right),\hfill & B=\left(\begin{array}{ll}0\hfill & 0\hfill \\ D^{-1}\hfill & 0\hfill \end{array}\right)\hfill \end{array}.$

Bounded solutions to (2.12), namely (ℒ - λ)U = 0, and (2.13) are then in one-to-one correspondence.

In particular, if Q(·) is not a constant function, then λ = 0 is an eigenvalue of ℒ with eigenfunction Q _ξ(ξ). This can be seen by taking the derivative of (2.11) with respect to ξ which gives

$D{(Q_{\xi })}_{\xi \xi }+c{(Q_{\xi })}_{\xi }+{\partial }_{U}F{(Q)}_{\xi }=0$

so that ℒQ _ξ. Hence, $u\left(\xi \right)=\left(Q_{\xi }\left(\xi \right),Q_{\xi \xi }\left(\xi \right)\right)$ satisfies (2.13) for λ = 0.

One important example is the FitzHugh-Nagumo equation (FHN)

$\begin{array}{l}{u}_t=u_{xx}+f(u)-w,\\ w_t={\delta }^2{w}_{xx}+\varepsilon (u-\gamma w),\end{array}$

for instance with f(u) = u(1 - u)(u - a). It admits various travelling waves such as pulses, fronts and backs (see, e.g., [91,105,176] for references). The stability of pulses has been studied in [90,185]. Stability results for spatially-periodic wave trains can be found in [53, 156], whereas the stability of concatenated fronts and backs has been studied in [124,147] and [125,154].

Many other results on the stability of waves to reaction-diffusion equations can be found in the literature (see, e.g., [47,60]). One class of such equations that has been studied extensively are monotone systems (see [37,141,182] and Section 8). We also refer to [85, Section 5.4] for instructive examples.

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Handbook of the Geometry of Banach Spaces

Apostolos A. Giannopoulos , Vitali D. Milman , in Handbook of the Geometry of Banach Spaces, 2001

1 Introduction

In this article we discuss results which stand between geometry, convex geometry, and functional analysis. We consider the family of n-dimensional normed spaces and study the asymptotic behavior of their parameters as the dimension n grows to infinity. Analogously, we study asymptotic phenomena for convex bodies in high dimensional spaces.

This theory grew out of functional analysis. In fact, it may be viewed as the most recent one among many examples of directions in mathematics which were born inside this field during the twentieth century. Functional analysis was developed during the period between the World Wars by the Polish school of mathematics, an outstanding school with broad interests and connections. The influence of the ideas of functional analysis on mathematical physics, on differential equations, but also on classical analysis, was enormous. The great achievements and successful applications to other fields led to the creation of new directions (among them, algebraic analysis, noncommutative geometry and the modern theory of partial differential equations) which in a short time became autonomous and independent fields of mathematics.

Thus, in the last decades of the twentieth century, geometric functional analysis and even more narrowly the study of the geometry of Banach spaces became the main line of research in what remained as "proper" functional analysis. The two central themes of this theory were infinite dimensional convex bodies and the linear structure of infinite dimensional normed spaces. Several questions in the direction of a structure theory for Banach spaces were asked and stayed open for many years. Some of them can be found in Banach's book. Their common feature was a search for simple building blocks inside an arbitrary Banach space. For example: does every Banach space contain an infinite unconditional basic sequence? Is every Banach space decomposable as a topological sum of two infinite dimensional subspaces? Is it true that every Banach space is isomorphic to its closed hyperplanes? Does every Banach space contain a subspace isomorphic to some ℓ_p or c ₀?

This last question was answered in the negative by Tsirelson (1974) who gave an example of a reflexive space not containing any ℓ_p . Before Tsirelson's example, it had been realized by the second named author that the notion of the spectrum of a uniformly continuous function on the unit sphere of a normed space was related to this question and that the problem of distortion was a central geometric question for approaching the linear structure of the space. Although Tsirelson's example was a major breakthrough and introduced a completely new construction of norm, the search for simple linear structure continued to be the aim of most of the efforts in the geometry of Banach spaces. We now know that infinite dimensional Banach spaces have much more complicated structure than what was assumed (or hoped). All the questions above were answered in the negative in the middle of the 90s, starting with the works of Gowers and Maurey, Gowers, Odell and Schlumprecht. Actually, the line of thought related to Tsirelson's example and the concepts of spectrum and distortion were the most crucial for the recent developments.

The systematic quantitative study of n-dimensional spaces with n tending to infinity started in the 60s, as an alternative approach to several unsolved problems of geometric functional analysis. This study led to a new and deep theory with many surprising consequences in both analysis and geometry. When viewed as part of functional analysis,this theory is often called local theory (or asymptotic theory of finite dimensional normed spaces). However, it adopted a significant part of classical convexity theory and used many of its methods and techniques. Classical geometric inequalities such as the Brunn–Minkowski inequality, isoperimetric inequalities and many others were extensively used and established themselves as important technical tools in the development of local theory. Conversely, the study of geometric problems from a functional analysis point of view enriched classical convexity with a new approach and a variety of problems: The "isometric" problems which were typical in convex geometry were replaced by "isomorphic" ones, with the emphasis on the role of the dimension. This change led to a new intuition and revealed new concepts, the concentration phenomenon being one of them, with many applications in convexity and discrete mathematics. This natural melting of the two theories should perhaps correctly be called asymptotic (or convex) geometric analysis.

This paper presents only some aspects of this asymptotic theory. We leave aside type-cotype theory and other connections with probability theory, factorization results, covering and entropy (besides a few results we are going to use), connections with infinite dimension theory, random normed spaces, and so on. Other articles in this collection will cover these topics and complement these omissions. On the other hand, we feel it is necessary to give some background on convex geometry: This is done in Sections 2 and 3.

The theory as we build it below "rotates" around different Euclidean structures associated with a given norm or convex body. This is in fact a reflection of different traces of hidden symmetries every high dimensional body possesses. To recover these symmetries is one of the goals of the theory. A new point which appears in this article is that all these Euclidean structures that are in use in local theory have precise geometric descriptions in terms of classical convexity theory: they may be viewed as "isotropic" ones.

Traditional local theory concentrates its attention on the study of the structure of the subspaces and quotient spaces of the original space (the "local structure" of the space). The connection with classical convexity goes through the translation of these results to a "global" language, that is, to equivalent statements pertaining to the structure of the whole body or space. Such a comparison of "local" and "global" results is very useful, opens a new dimension in our study and will lead our presentation throughout the paper.

We refer the reader to the books of Schneider [177] and of Burago and Zalgaller [43] for the classical convexity theory. Books mainly devoted to the local theory are the ones by: Milman and Schechtman [150], Pisier [164], Tomczak-Jaegermann [195].

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Uniform Spaces

Eric Schechter , in Handbook of Analysis and Its Foundations, 1997

UNIFORM CONTINUITY

18.7.

Notation. Let u be a uniform space, with uniformity u determined by gauge D. Let ((x _α, x′_α) : α ∈ A) be a net in X×X, and let ε be its eventuality filter on X×X.

Show that the following conditions are equivalent.

(A)

u ⊆ ∈

(B)

For each U ∈ u, we have eventually (x _α, x′_α) ∈ U.

(C)

d(x _α, x′_α) → 0 in ℝ for each d ∈ D. We shall abbreviate this as D(x _α, x′_α) → 0.

We emphasize that the last condition does not say ${\sup }_{d\in D}d\left(x_{\alpha },{x^{\prime }}_{\alpha }\right)\to 0.$

18.8.

Definition. Let X,U and (Y, v) be uniform spaces, and let D and E be any gauges that determine the uniformities U and v, respectively. Let φ : X→ Y be some function. Then the following conditions are equivalent. If any (hence all) of these conditions hold, we say φ is uniformly continuous.

(A)

Whenever V∈ v, then the set

$\begin{array}{lll}{\left(\phi \times \phi \right)}^{-1}\left(V\right)\hfill & =\hfill & \left\{\begin{array}{lll}\left(x,x^{\prime }\right)\in X\times X\hfill & :\hfill & \left(\phi \left(x\right),\phi \left(x^{\prime }\right)\right)\in V\hfill \end{array}\right\}\hfill \end{array}$

is a member of U. That is, the inverse image of each entourage is an entourage. (This is the definition of uniform continuity used in 9.8.)

(B)

Whenever D(x _α, x′_α) → 0 in X, then E(φ(x _α), φ(x″_α)) → 0 in Y. (Notation is as in 18.7(C).)

(C)

For each number ε > 0 and each pseudometric e∈ E, there exist some number δ > 0 and some finite set D′ ⊆ D such that

$\begin{array}{lll}\underset{d\in D^{\prime }}{\max }d\left(x_1,x_2\right)<\delta \hfill & \Rightarrow \hfill & e\left(\phi \left(x_1\right),\phi \left(x_2\right)\right)<\epsilon .\hfill \end{array}$

(We emphasize that the choice of δ and D′ depends on ε and e but not on x ₁ or x ₂. Compare this with 15.14(D).)

(D)

For each e∈ E, there exists a finite set D _e ⊆ D and a function r _e : [0, +∞) → [0,+∞) that is continuous and increasing, and satisfies γ _e (0) = 0 and

$\begin{array}{lll}e\left(\phi \left(x\right),\phi \left(x^{\prime }\right)\right)\hfill & \le \hfill & {\gamma }_e\left(\underset{d\in D_e}{\max }d\left(x,x^{\prime }\right)\right)\hfill \end{array}.$

Such a system of sets D _e and functions γ _e will be called a modulus of uniform continuity for φ.

Note that if the gauge D is directed (as defined in 4.4.c), then conditions 18.8(C) and 18.8(D) can be simplified slightly: the sets D' and D _c may be taken to be singletons {d}.

18.9.

Examples and related properties

a.

If the uniformity on X is given by a pseudometric d, then sequences suffice in 18.8(B) (regardless of whether Y is pseudometrizable). That is, a mapping φ : X→ Y is uniformly continuous if and only if

$\text{whenever}d\left(x_n,{x^{\prime }}_n\right)\to 0\text{inX, then}E\left(\phi \left(x_n\right),\phi \left({x^{\prime }}_n\right)\right)\to 0\text{ ;in}Y,$

with notation as in 18.7(C).

b.

Any Hölder-continuous function from one metric space into another is uniformly continuous. The converse is false. For instance, define $f:\left[0,e^{-1}\right]\to ℝ$

$\begin{array}{lll}f\left(t\right)\hfill & =\hfill & \{\begin{array}{cc}0& \text{when}t=0\\ -1/\ln t& \text{when}0<t\le e^{-1}\end{array}\hfill \end{array}$

Show that f is not Hölder continuous with any exponent. It is easy to see that f is continuous; then the uniform continuity of f will follow by a compactness argument in 18.21.

c.

Any uniformly continuous function is continuous (where each uniform space is equipped with its uniform topology). This can be proved using uniformities or using gauges; the student is urged to give both proofs.

d.

Show that the function f(t) = 1/t is continuous, but not uniformly continuous, on the open interval (0, 1). Use this fact to give two different metrics on (0, 1) that yield different uniformities but that both yield the usual topology.

e.

(Preview.) Let p: X→ Y be a linear map from one topological vector space to another — or more generally, an additive map from one topological Abelian group to another. Let X and Y be equipped with their usual uniform structures (see 26.37). If p is continuous, then p is uniformly continuous; see 26.36.c.

f.

Let X be a set, let {(Y _λ, E _λ) : λ ∈ ∧} be a collection of gauge spaces, and let φ_λ : X→ Y _λ be some mappings. Show that the initial uniformity on X determined by the φ_λ's and E _λ's (as in 9.16) is equal to the uniformity on X determined by the gauge $D={\displaystyle {\cup }_{\lambda \in \wedge }\left\{e{\phi }_{\lambda }:e\in E_{\lambda }\right\}}$ where $\left(e{\phi }_{\lambda }\right)\left(x,x^{\prime }\right)=e\left({\phi }_{\lambda }\left(x\right),{\phi }_{\lambda }\left(x^{\prime }\right)\right).$ We may call this the initial gauge determined by the φ_λ's and E _λ's (although any other gauge uniformly equivalent to this one will generally do just as well).

An important special case: When $X={\displaystyle {\prod }_{\lambda \in \wedge }{Y}_{\lambda }}$ and the φ_λ's are the coordinate projections, we obtain the product gauge.

g.

The forgetful functor from uniform spaces to topological spaces preserves the formation of initial objects.

That is, the uniform topology (U) determined by an initial uniformity πλ S determined by π_λ's and uniformities v _λ is equal to the initial topology determined by the λ_τ's and the uniform topologies τ(v _λ) determined by those uniformities.

18.10.

Theorem on uniform continuity of extensions. Let X and Y be uniform spaces, let X ₀ ⊆ X be dense, let φ : X→ Y be continuous, and suppose that the restriction of φ to X ₀ is uniformly continuous. Then φ is uniformly continuous on X. In fact, if some gauges are specified for X and Y, then any modulus of uniform continuity for the restriction of φ to X ₀ is also a modulus of uniform continuity for φ on X.

In particular, if φ is continuous and the restriction of φ to X ₀ is Hülder continuous or Lipschitzian, then φ is HüUlder continuous or Lipschitzian with the same constant.

Hints: Use notation as in 18.8(D). Let any x, x′ ∈ X be given. Choose a net ((x _α,x′_α)) in X ₀ × X ₀ that converges to (x, x′). For each α, we have

$\begin{array}{ccc}e\left(\phi \left(x_{\alpha }\right),\phi \left({x^{\prime }}_{\alpha }\right)\right)& \le & \gamma e\left(\underset{d\in D_e}{\max }d\left(x_{\alpha },{x^{\prime }}_{\alpha }\right)\right)\end{array}.$

Holding e fixed, take limits to obtain a corresponding inequality for (x, x′).

18.11.

Characterization of uniformly equivalent gauges. Let D and E be gauges on a set X. Then the following conditions are equivalent:

(A)

D and E are uniformly equivalent — i.e., they generate the same uniformity.

(B)

The identity map i _X : X→ X is uniformly continuous in both directions between the gauge spaces (X, D) and (X, E)

(C)

For each net ((x _α,x′_α) : α ∈ A) in X×X, we have

$\begin{array}{ccc}D\left(x_{\alpha },{x^{\prime }}_{\alpha }\right)\to 0& \iff & E\left(x_{\alpha },{x^{\prime }}_{\alpha }\right)\to 0\end{array}$

with notation as in 18.7(C).

Hint: A uniformity, being a proper filter, is the eventuality filter for some net.

18.12.

Further exercise. Let U be a uniformity on a set X. Then the largest gauge that is compatible with U (as defined in 5.32) is the set of all pseudometrics d: X×X→ [0, +∞) that are jointly uniformly continuous — i.e., uniformly continuous when X×X is given its product uniformity and [0, +∞) is given its usual uniformity. (Compare this with 16.20.)

18.13.

If D is any gauge, then D is uniformly equivalent to its max closure and its sum closure, defined as in 4.4.c.

(Hence it is often possible to replace a gauge with a directed gauge; thus in many contexts we may assume a gauge is directed.)

18.14.

Definition. We shall say β is a bounded remetrization function if:

(i)

β is a mapping from [0, +∞) onto a bounded subset of [0, +∞);

(ii)

β is continuous;

(iii)

β is increasing; that is, s≤ t⇒ β(s) ≤ β(t);

(iv)

β(t) = 0 ⇔ t= 0; and

(v)

β is subadditive; that is, β(s + t) ≤ β(s) + β(t).

Show that

a.

arctan(t), tanh(t), min{1, t}, and t/(1 + t) are bounded remetrization functions of t. (Hint: See 12.25.e.) Note that min{1, t} is not injective.

b.

If β is a bounded remetrization function and d is a (pseudo)metric on a set X, then e(x, y) = β(d(x, y)) defines a (pseudo)metric e= β оd on X that is uniformly equivalent to d and is bounded.

c.

If β is a bounded remetrization function and D is a gauge on a set X, then {β о d: d∈ D} is a gauge on X that is uniformly equivalent to D and is uniformly bounded — i.e., we have sup{β(d(x, y)) : x, y ∈ X, d ∈ D} < ∞.

18.15.

Example. The usual metric on ℝ is d(x, y) = |x - y|. Another metric, bounded and uniformly equivalent to the usual one, is e(x, y) = arctan(|x- y|). On the other hand, ρ(x, y) = | arctan(x) – arctan(y)| is a bounded metric on ℝ that is equivalent, but not uniformly equivalent, to the usual metric. (All three metrics yield the same topology.)

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