# Questions tagged [function-spaces]

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55
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### "Calculus" of metric entropy: How does metric entropy behave with respect to binary operators?

This is a general question, more of a reference request. tl;dr: Is there a "calculus" for computing metric entropy bounds?
Given a function space $\mathcal{F}$, we may define its metric ...

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2
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229
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### What is the relationship between Hölder spaces and differentiability?

I'm porting this question over from MSE as it did not get any responses other than one comment on there.
Let $C^{k,\alpha}$ be a Hölder space where $0 \leq \alpha \leq 1$. I have seen various sources ...

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### A generalization of polynomials in one variable

Let us consider the space of polynomials $P^N$ of degree $\le N$. If $f\in P^N$ vanishes in $>N$ points, then $f\equiv 0$, but for any $N$ points, or fewer, there exists $f\neq 0$ vanishing at ...

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### Why do we work on homogeneous Besov/Triebel-Lizorkin spaces?

This question is mainly for understanding the history behind homogeneous spaces.
There is extensive literature on Besov and Triebel-Lizorkin spaces. For instance, see the standard textbook:
https://...

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### Is $T$ totally bounded when $C_u(T)$ is separable?

I'm seeking help with a question regarding the space of bounded and uniformly continuous functions $C_u(T,d)$, where $(T,d)$ is a metric space. In this context, $C_u(T)$ is a closed subspace of $C_b(T)...

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### Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?

The following startles me. Let $\Omega \subseteq \mathbb R^n$ and write $W^{\sigma,1}(\Omega)$ for the fractional Sobolev space with norm
$$|u|_{W^{\sigma,1}(\Omega)} := \iint \frac{|u(x) - u(y)|}{|x-...

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### The graph topologies for powersets

Given a topological space $X$ and a metric space $(Y,d_Y)$, there are a number of topologies one may put on the space $\mathcal{C}(X,Y)$ of continuous functions from $X$ to $Y$. Perhaps the most ...

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### Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?

Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...

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### What happens if we consider functions of bounded variation that are not in $L^1$?

A function $f \in L^1(\mathbb R^n)$ is said to be of bounded variation if there exists a constant $C \geq 0$ such that
$$
\int_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx
\leq
C \sup_{ x \in \...

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### Are algebras of smooth functions formally smooth?

Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$?
If it helps, feel free to assume that $M$ is compact.
(This is not a joke ...

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### Interpolation between (or: simultaneous Whitney extension for) $C^\alpha$ and $C^{1,\gamma}$ on a Lipschitz domain

I would like to know whether for a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ (in the weak Lipschitz, so a "Lipschitz manifold", sense, not necessarily a Lipschitz graph domain), ...

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### What is t-equivalence in function spaces?

In $C_p$-Theory monographs, it is said that two topological spaces $X$ and $Y$ are said to be $t$-equivalent means that $C_p(X)$ is homeomorphic to $C_p(Y)$. Then they also define $u$-equivalences (...

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### quasi-Banach function spaces are subspace of $L_p$

It is well-known that any Banach rearrangement-invariant function space $X$ on $[0,1]$ is a subset of $L_1[0,1]$, and I can find a reference that any quasi-Banach rearrangement-invariant function ...

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### Relationship between Hardy-Orlicz space and the corresponding Orlicz space

For $p \in [1, \infty]$ the Hardy space $H_p$ is defined as the space of all analytic functions $f$ on the open disk satisfying
$$\|f\|_{H_p} = \sup_{0 < r < 1} \|f(r\cdot)\|_{L_p(\mathbb{T})} &...

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### Schauder basis of $L^1_{\mathrm{loc}}(\mathbb{R}^n,H)$

$\newcommand{\loc}{\mathrm{loc}}$Let $(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n),\mu)$ denote the Euclidean space $\mathbb{R}^n$ with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ equipped with ...

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326
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### About the normability of the space of continuous functions

Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that ...

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### Relationship between $C(X\times Y,Z)$ and $C(X,C(Y,Z))$

Let $X$, $Y$, and $Z$ be locally-compact, complete, and separable metric spaces and suppose that $X$ is compact; all non-empty.
Consider the spaces $C(X,C(Y,Z))$ and $C(X\times Y,Z)$ both equipped ...

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### Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?

I know that $\ell_2$ is isomorphic to a subspace of $L_p(0,1)$ for any $1\le p<\infty$. However, I haven't seen anything about $L_\infty$. Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?...

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### Determining a space of differentiability

I have a questions and maybe you are able to assist with this? Let us consider the space $X:=\mathrm{L}^2[0,\pi]$. On $X$ we consider the family of operators $(P(t,s))_{t\geq s}$ defined by
$$
P(t,s)f:...

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### Topologies on space of compactly supported continuous functions

Let $X$ be a locally compact Hausdorff space. As far as I understand, the space $C_c(X) = C_c(X; \mathbb{C})$ of compactly supported continuous complex-valued functions on $X$ is (most?) often ...

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### Has this Banach algebra been studied?

Given $\Omega$ as $[0,1]^n$ or the closed unit ball in $\mathbb{R}^n$, we can consider the algebra of complex valued polynomials with pointwise multiplication and its closure with respect to the norm
...

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53
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### Where can I find literature regarding cardinal invariants of a function space $C(X, Y)$ endowed with the Uniform or Fine topology?

I am working on Function Spaces as a topological space. I want to get a sample paper which studies the cardinal invariants on the function space $C(X, Y)$ rather than on $C(X)$.

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### What is the source to find cardinal invariants for a function space C(X, Y), equipped with uniform or fine topology?

I would like to know about the technique to check the cardinality properties for the function space C(X, Y), where X is a tychonoff space and Y a metric space, equipped with uniform or fine topology.

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235
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### Which domains have a Poincare-Wirtinger inequality? Which don't?

A Poincare-Wirtinger inequality holds over a domain $\Omega \subseteq \mathbb R^n$ with exponentnt $1 \leq p \leq \infty$ if there exists $C(p,\Omega) > 0$ such that
$\| u - \operatorname{avg}(u) \|...

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### Can we define geodesic in the space of compactly supported functions?

From Wikepedia, the definition of geodesic is stated as:
A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...

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### strong topologies on $C_c^\infty$

UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...

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295
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### Analogue of Lipschitz continuity of $W^{1,\infty}$ for Hölder continuity and Sobolev-Slobodeckij spaces

A function $u : U \rightarrow \mathbb R$ is an element of the Hölder space $C^{\alpha}(U)$ if
$\sup\limits_{x \in U} |u(x)| < \infty$
$\sup\limits_{x,y \in U} \dfrac{|u(x) - u(y)|}{|x-y|^\alpha} &...

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### On $B^1$ and $B^2$ almost-periodic functions

The Besicovitch class of $B^p$ almost-periodic functions is defined as the closure of the set of trigonometric polynomials (of the form $t \mapsto \sum_{n=1}^N a_n e^{i \lambda_n t}$ with $\lambda_1, \...

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### The set of embeddings is open in the strong Whitney topology

In Hirsch's book "Differential Topology," he claims in Chapter 2, Theorem 1.4 that the set of $C^1$-embeddings is open in the strong Whitney topology $C^1(M, N)$ where $M$ and $N$ are $C^1$ manifolds. ...

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345
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### Topological spaces containing paths

Let $C(\mathbb{R}^n;\mathbb{R}^d)$ be the space of continuous functions with the uniform-convergence on compacts topology. What are function spaces $X$ building on $C(\mathbb{R}^n;\mathbb{R}^d)$?
$X$...

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### Which set of functions/measures has range $\mathrm{L}^\infty$ under Fourier transformation

I have a question concerning the Fourier transformation. What I know is that $\mathrm{L}^{\infty}=\{\hat{u}:\ u\in Y\}$ for some space $Y$. Now, I want to specify the space $Y$. The question is, is ...

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### Uniform convergence over compacts subsets implies existence of a uiform convergente subsequence?

Let $H$ the group of all homeomorphisms of a locally compact second countable and totally bounded metric space $X$ onto itself, under the compact-open topology ($X$ is totally bounded if every ...

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353
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### Criteria for $\epsilon$-Density

Let $Y$ be a compact, separable metric space and $X=C(Y)$ Banach space. There are many criteria for a linear subspace $Z\subseteq X$ to be dense; notably the Stone-Weierstraß theorem.
Are there ...

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### Is $π:\mathcal{C}^∞(M,N)→\mathcal{C}^∞(S,N)$, $π(f)=f|_S$ a quotient map in the $\mathcal{C}^1$ topology?

This question was previously posted on MSE.
Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)$...

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### The regularity of ODE with Zygmund coefficients

A zygmund function $f\in\mathscr C^1$ is a continuous function satisfies $|f(x+h)+f(x-h)-2f(x)|\le C|h|$ for all $x,h\in\mathbb R^n$ in the domain.
According to Markus' paper A uniqueness theorem for ...

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### Notions of $\beta$-Hölder smoothness when $\beta\in (1,2]$: are they equivalent?

I posted the following question on StackExchange a few months ago (https://math.stackexchange.com/questions/2898620/notions-of-beta-h%C3%B6lder-smoothness-when-beta-in-1-2-are-they-equivalent), but ...

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### Reference request: interpolation of Hölder spaces

On the Wikipedia page on interpolation space, it is written that the space $C^\theta([0, 1])$ is the (real) interpolation of $C^0([0, 1])$ and $C^1([0, 1])$, where $C^\theta([0, 1])$ denotes the space ...

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### Logarithm of $L^p$ space

I encountered the following space as a natural space for setting up a certain problem:
$$
S_m^p = \{f \colon I \to \mathbb{R} \text{ measurable }; m^{f} \in L^p(I)\}
$$
Here, $I$ is an open bounded ...

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0
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### Pointwise convergence in Lawvere metric spaces

In the formalism of Lawvere metric spaces, we have that the distance in the hom-space $[X,Y]$ is given by:
$$
d(f,g) = \sup_{x\in X} d(f(x),g(x)) .
$$
Therefore, a sequence of functions $f_n:X\to Y$ ...

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votes

1
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194
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### Recognizing locally convex spaces on which all bounded linear functionals are continuous

Is it possible to characterize the Hausdorff locally convex spaces on which all bounded linear functionals are continuous?
It is known that a space is bornological if and only if the space is Mackey ...

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### When is the strict topology bornological?

Let $X$ be a completely regular Hausdorff space. Are there known conditions under which the algebra of bounded continuous functions on $X$, endowed with the strict topology, is bornological?
(Of ...

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### Working in coordinates with topologies on the algebra of continuous functions

Let $X$ be a Hausdorff completely regular topological space, and let $C_b (X)$ be its algebra of continuous bounded functions. Endow $C_b (X)$ with a topology given by some seminorms, that contains ...

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### "Reversion" of class $J(\theta)$ interpolation property for Besov spaces

In (function space) interpolation theory, a Banach space $E$ is of class $J(\theta)$ (for $0 < \theta < 1$) if $$X \cap Y \subseteq E \subseteq X+Y,$$ where $(X,Y)$ are Banach spaces and form an ...

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434
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### Characterizing the sum $L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1}$ of iterated Lebesgue spaces "by duality"

For the usual Lebesgue spaces $L^p (\mu)$ ($p \in [1,\infty]$) on a ($\sigma$-finite) measure space $(X,\mu)$, it is well-known that one has the characterization
$$
L^p (\mu) = \left\{f : X \to \Bbb{...

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149
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### Has anybody studied continued fractions in function spaces?

For the text below, define $f^\infty(x) = \lim_{n\to\infty} f^n(x)$ where $f^n = \underbrace{f \circ \ldots \circ f}_{n}$.
Usually 'continued fraction' means continued fraction in $\mathbb{R}$. For ...

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### The topological duals of spaces of finite measures

In volume 1 of "Linear Operators", Dunford and Schwartz say that (footnote F1, page 374)
"No completely satisfactory representation for the conjugate space of $ba(S, \Sigma)$, $ca(S, \Sigma)$ or $...

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### Imbedding Theorems between Besov Spaces and space of continuos functions on the unit circle

I'll try to be brief.
Let us consider the Besov Space $B^{1/p}_{p, p}(\mathbb{T})$, where $1\leq p<\infty $ and $\mathbb{T}$ is the unit circle in the complex plane. I would like to know for which ...

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### Wavelet characterization of Sobolev spaces

We know that there exist wavelets generating orthonormal bases in Sobolev spaces $W^{p,s}(\mathbb R^n)$, where $p$ is the index of integral and $s$ is the index of smoothness. Consider the orthonormal ...

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### Density of certain rational functions in the Hilbert space $L^2(-\infty,0)$

It is easy to check that the functions
$$f_{n,z}(x):=(z-x)^{-n},\quad n\geq 1,\quad z\in \mathbb{C}-(-\infty,0]$$ belong to the Hilbert space $L^2(-\infty,0)$, i.e., $L^2$-integrable complex-valued ...

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### Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense:
A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...