Questions tagged [function-spaces]
The function-spaces tag has no usage guidance.
55
questions
0
votes
0
answers
47
views
"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 ...
- 9
2
votes
2
answers
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 ...
- 379
1
vote
0
answers
148
views
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 ...
- 398
2
votes
0
answers
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://...
4
votes
1
answer
178
views
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)...
- 91
1
vote
1
answer
82
views
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
$$|u|_{W^{\sigma,1}(\Omega)} := \iint \frac{|u(x) - u(y)|}{|x-...
- 4,706
2
votes
0
answers
54
views
The graph topologies for powersets
Given a topological space $X$ and a metric space $(Y,d_Y)$, there are a number of topologies one may put on the space $\mathcal{C}(X,Y)$ of continuous functions from $X$ to $Y$. Perhaps the most ...
- 9,697
0
votes
0
answers
134
views
Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1
7
votes
2
answers
499
views
What happens if we consider functions of bounded variation that are not in $L^1$?
A function $f \in L^1(\mathbb R^n)$ is said to be of bounded variation if there exists a constant $C \geq 0$ such that
$$
\int_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx
\leq
C \sup_{ x \in \...
- 4,706
12
votes
1
answer
361
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 ...
- 5,596
5
votes
0
answers
169
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), ...
- 2,100
1
vote
0
answers
82
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 (...
- 31
2
votes
0
answers
71
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 ...
- 597
3
votes
0
answers
54
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
$$\|f\|_{H_p} = \sup_{0 < r < 1} \|f(r\cdot)\|_{L_p(\mathbb{T})} &...
- 493
3
votes
1
answer
303
views
Schauder basis of $L^1_{\mathrm{loc}}(\mathbb{R}^n,H)$
$\newcommand{\loc}{\mathrm{loc}}$Let $(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n),\mu)$ denote the Euclidean space $\mathbb{R}^n$ with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ equipped with ...
- 5,001
0
votes
1
answer
326
views
About the normability of the space of continuous functions
Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that ...
- 977
2
votes
1
answer
177
views
Relationship between $C(X\times Y,Z)$ and $C(X,C(Y,Z))$
Let $X$, $Y$, and $Z$ be locally-compact, complete, and separable metric spaces and suppose that $X$ is compact; all non-empty.
Consider the spaces $C(X,C(Y,Z))$ and $C(X\times Y,Z)$ both equipped ...
0
votes
0
answers
140
views
Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?
I know that $\ell_2$ is isomorphic to a subspace of $L_p(0,1)$ for any $1\le p<\infty$. However, I haven't seen anything about $L_\infty$. Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?...
- 597
2
votes
0
answers
49
views
Determining a space of differentiability
I have a questions and maybe you are able to assist with this? Let us consider the space $X:=\mathrm{L}^2[0,\pi]$. On $X$ we consider the family of operators $(P(t,s))_{t\geq s}$ defined by
$$
P(t,s)f:...
3
votes
2
answers
753
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 ...
- 89
7
votes
0
answers
238
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
...
- 71
2
votes
0
answers
53
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 $C(X)$.
- 31
0
votes
1
answer
69
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
235
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) \|...
- 4,706
2
votes
1
answer
150
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 ...
- 163
14
votes
0
answers
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 "...
- 476
2
votes
0
answers
295
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} &...
- 4,706
1
vote
1
answer
93
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, \...
- 1,144
6
votes
2
answers
487
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. ...
- 1,152
2
votes
1
answer
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 $C(\mathbb{R}^n;\mathbb{R}^d)$?
$X$...
- 5,001
1
vote
0
answers
80
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
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 ...
- 355
0
votes
1
answer
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 ...
- 5,001
6
votes
1
answer
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 $\mathcal C^\infty(M, N)$...
- 233
2
votes
0
answers
206
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 ...
- 786
3
votes
0
answers
88
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 ...
- 31
3
votes
1
answer
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 ...
- 1,223
2
votes
0
answers
98
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 ...
- 628
1
vote
0
answers
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 $f_n:X\to Y$ ...
- 2,099
3
votes
1
answer
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 ...
- 5,105
1
vote
1
answer
136
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 ...
- 5,105
1
vote
0
answers
66
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 ...
- 5,105
7
votes
2
answers
504
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 ...
- 2,100
7
votes
0
answers
434
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
$$
L^p (\mu) = \left\{f : X \to \Bbb{...
- 754
2
votes
0
answers
149
views
Has anybody studied continued fractions in function spaces?
For the text below, define $f^\infty(x) = \lim_{n\to\infty} f^n(x)$ where $f^n = \underbrace{f \circ \ldots \circ f}_{n}$.
Usually 'continued fraction' means continued fraction in $\mathbb{R}$. For ...
- 29
5
votes
1
answer
575
views
The topological duals of spaces of finite measures
In volume 1 of "Linear Operators", Dunford and Schwartz say that (footnote F1, page 374)
"No completely satisfactory representation for the conjugate space of $ba(S, \Sigma)$, $ca(S, \Sigma)$ or $...
- 5,105
2
votes
0
answers
116
views
Imbedding Theorems between Besov Spaces and space of continuos functions on the unit circle
I'll try to be brief.
Let us consider the Besov Space $B^{1/p}_{p, p}(\mathbb{T})$, where $1\leq p<\infty $ and $\mathbb{T}$ is the unit circle in the complex plane. I would like to know for which ...
- 141
3
votes
0
answers
213
views
Wavelet characterization of Sobolev spaces
We know that there exist wavelets generating orthonormal bases in Sobolev spaces $W^{p,s}(\mathbb R^n)$, where $p$ is the index of integral and $s$ is the index of smoothness. Consider the orthonormal ...
- 231
1
vote
1
answer
160
views
Density of certain rational functions in the Hilbert space $L^2(-\infty,0)$
It is easy to check that the functions
$$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 ...
- 11
7
votes
1
answer
369
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 ...
- 141