# Questions tagged [function-spaces]

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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 ...
229 views

### 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 ...
1 vote
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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 ...
66 views

### 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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### 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 ...
605 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 "...
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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. ...
345 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 $C(\mathbb{R}^n;\mathbb{R}^d)$? $X$...
1 vote
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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 ...
1 vote
172 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 ...
353 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 ...
295 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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### 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 ...
866 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 ...
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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 ...
1 vote
97 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 $f_n:X\to Y$ ...
194 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 ...
1 vote
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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 ...
1 vote
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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 ...
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 $$L^p (\mu) = \left\{f : X \to \Bbb{... 2 votes 0 answers 149 views ### Has anybody studied continued fractions in function spaces? For the text below, define f^\infty(x) = \lim_{n\to\infty} f^n(x) where f^n = \underbrace{f \circ \ldots \circ f}_{n}. Usually 'continued fraction' means continued fraction in \mathbb{R}. For ... 5 votes 1 answer 575 views ### The topological duals of spaces of finite measures In volume 1 of "Linear Operators", Dunford and Schwartz say that (footnote F1, page 374) "No completely satisfactory representation for the conjugate space of ba(S, \Sigma), ca(S, \Sigma) or ... 2 votes 0 answers 116 views ### Imbedding Theorems between Besov Spaces and space of continuos functions on the unit circle I'll try to be brief. Let us consider the Besov Space B^{1/p}_{p, p}(\mathbb{T}), where 1\leq p<\infty  and \mathbb{T} is the unit circle in the complex plane. I would like to know for which ... 3 votes 0 answers 213 views ### Wavelet characterization of Sobolev spaces We know that there exist wavelets generating orthonormal bases in Sobolev spaces W^{p,s}(\mathbb R^n), where p is the index of integral and s is the index of smoothness. Consider the orthonormal ... 1 vote 1 answer 160 views ### Density of certain rational functions in the Hilbert space L^2(-\infty,0) It is easy to check that the functions$$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 ...
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 ...