Questions tagged [function-spaces]

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12 votes
1 answer
293 views

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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0 votes
0 answers
46 views

Equivalent formulation of uniform convergence in measure

If $(X,m)$ is a measure space, and $f_n, f : X \to \mathbb C$ are measurable functions, then it is known that $f_n \to f$ in measure if and only if every sub-sequence $(f_{n_i}) _{i \ge 0}$ contains a ...
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5 votes
0 answers
133 views

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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1 vote
0 answers
69 views

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 (...
2 votes
0 answers
57 views

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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2 votes
0 answers
40 views

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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3 votes
1 answer
197 views

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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0 votes
1 answer
179 views

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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2 votes
1 answer
167 views

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 ...
0 votes
0 answers
128 views

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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2 votes
0 answers
48 views

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:...
2 votes
2 answers
412 views

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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7 votes
0 answers
234 views

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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2 votes
0 answers
42 views

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 $C(X)$.
0 votes
1 answer
63 views

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.
2 votes
0 answers
153 views

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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2 votes
1 answer
120 views

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 ...
14 votes
0 answers
483 views

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 "...
2 votes
0 answers
197 views

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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1 vote
1 answer
82 views

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, \...
6 votes
2 answers
370 views

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. ...
2 votes
1 answer
336 views

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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1 vote
0 answers
71 views

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 ...
1 vote
0 answers
137 views

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 ...
0 votes
1 answer
284 views

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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6 votes
1 answer
256 views

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 $\mathcal C^\infty(M, N)$...
2 votes
0 answers
192 views

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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3 votes
0 answers
81 views

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 ...
3 votes
1 answer
534 views

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 ...
2 votes
0 answers
78 views

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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1 vote
0 answers
91 views

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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3 votes
1 answer
173 views

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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1 vote
1 answer
122 views

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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1 vote
0 answers
64 views

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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7 votes
2 answers
469 views

"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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7 votes
0 answers
373 views

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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2 votes
0 answers
146 views

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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5 votes
1 answer
552 views

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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2 votes
0 answers
106 views

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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3 votes
0 answers
195 views

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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1 vote
1 answer
121 views

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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7 votes
1 answer
350 views

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 ...
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2 votes
2 answers
292 views

Interpolation between $L_p$ and $B^s_{q,q}$

I am looking for a reference or a direct argument that shows the real interpolation space between $L_p$ and $B^s_{q,q}$ is $B^\alpha_{r,r}$, with the usual conditions on the indices. This result is ...
  • 3,252
5 votes
0 answers
493 views

Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is \begin{equation} \omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})} \end{equation} where $\Omega_{rh}=\{...
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24 votes
4 answers
6k views

Compact open topology

What is the intuition behind using compact open topology for eg. in the case of Pontryagin dual ?
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1 vote
1 answer
366 views

Continuous embedding of Hardy space in Lebesgue space

I would like to have a reference to the following statement which I think is true: $$h^1 \hookrightarrow L^1.$$ The closest I came to this is in D. Goldberg's paper, "A local version of real Hardy ...
4 votes
3 answers
2k views

Connected components of space of maps between two manifolds

Question: What are the connected components of the familiar spaces of functions between two (let's say compact and smooth, for simplicity) manifolds $M$ and $N$? Specifically, I'm thinking of the ...
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