# Questions tagged [function-spaces]

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28
questions

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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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**2**answers

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. ...

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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 ...

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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 ...

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**1**answer

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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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 ...

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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 ...

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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 ...

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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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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 $f_n:X\to Y$ ...

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**1**answer

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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**1**answer

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 ...

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**1**answer

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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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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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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**1**answer

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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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 ...

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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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**1**answer

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 ...

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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 ...

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**2**answers

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 ...

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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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**4**answers

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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**1**answer

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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**3**answers

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 ...