Questions tagged [function-spaces]

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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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2answers
177 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. ...
5
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1answer
327 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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67 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 ...
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88 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 ...
1
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1answer
106 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 ...
6
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1answer
177 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 $\mathcal C^\infty(M, N)$...
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0answers
155 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 ...
3
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0answers
74 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 ...
2
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1answer
140 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
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0answers
69 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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0answers
75 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 $f_n:X\to Y$ ...
3
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1answer
156 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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1answer
102 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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62 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 ...
7
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1answer
303 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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0answers
234 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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0answers
141 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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1answer
384 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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0answers
96 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 ...
3
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0answers
180 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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1answer
76 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 ...
7
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1answer
325 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 ...
3
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2answers
252 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 ...
6
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0answers
436 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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4answers
4k 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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1answer
354 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
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3answers
1k 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 ...