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A two variable real function $F(y,x)$, defined on $[c,d] \times [a,b]$, can be thought of as a linear map between functions of different domains by: $$ g(y) = \int^a_bF(y,x)f(x)dx $$ This has very ...
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Let $X$ and $Y$ and $Z$ be compactly generated Hausdorff spaces. Is it true that the composition map $Z^Y \times Y^X \rightarrow Z^X$ is continuous? Here the function spaces are given the compact-...
Some random guy's user avatar
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In the paper Wasserstein GANs are Minimax Optimal Distribution Estimators [1], the authors state within Lemma 3.3, p.12, that Besov spaces [2] of generalized smoothness (i.e., with the added parameter ...
feltshire's user avatar
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Let $1<p<\infty$ and $w\in A_p$ a weight in the Muckenhoupt class. In [Lemma 2.2] Fröhlich, A. The Stokes Operator in Weighted -Spaces I: Weighted Estimates for the Stokes Resolvent Problem in a ...
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Consider the $J$ sequence space and $JT$ the James tree space. Both are separable non-reflexive Banach spaces containing no isomorphic copies of $\ell_1$. Moreover, $JT$'s dual is non-separable, ...
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In 1967 Eliasson published his Article about geometry of manifolds of maps, in which he proposes a so called manifold model. He states roughly that a section functor $\mathcal F:Vect(M)\rightarrow Ban$...
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Let $T:L^p_{\mathrm{loc}}(\mathbb R)\to C_0(\mathbb R)$ be a bounded linear operator where $p<\infty$ is a suitable large number. Then we have boundedness of the tensor extension $I\otimes T:L^p_{\...
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Consider the Orlicz spaces $L^A(\mathbb{R}^n)$. Under suitable conditions for the $N$-function $A(x,t)$, namely $(Inc)_p$, $(Dec)_q,$, for $1<p<q<\infty$, it is known that $$L^p\cap L^q \...
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Assume every function is eventually nonnegative. In other words, we are interested in growth rates for measuring time complexity and such. $f = O(g)$ is equivalent to $\limsup \frac{f}{g} < \infty$,...
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It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
Guillermo García Sáez's user avatar
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Let $s\in (0,1)$ and $1<p<\infty$. Let $H^{s,p}(\mathbb{R}^n)=H^{s,p}$ the Bessel potential space, defined as the image of $L^p(\mathbb{R^n})$ by the Bessel potential. It is known that these ...
Guillermo García Sáez's user avatar
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I'm trying to prove that certain space, which can be obtained as an interpolation space, is separable. The fact that is separable is well known but i want to simplify it via interpolation. I haven't ...
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I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here. Let $C^k(\mathbb{R}^n$) be the ...
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Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
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Let $G \subseteq GL_d (\mathbb C)$ be a finite pseudoreflection group (see here and here) acting on a domain $\Omega \subseteq \mathbb C^d$ by the right action $\sigma \cdot z = \sigma^{-1} z$ where $\...
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Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with $$ [f]^{p}_{\dot{F}^{s}_{p,q}...
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Is the following true? A function $m:\mathbb R^n\to\mathbb C$ is a Schwartz multiplier (i.e. $[f\mapsto mf]:S(\mathbb R^n)\to S(\mathbb R^n)$ is bounded linear) iff the following: For every $\alpha$ ...
Liding Yao's user avatar
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Is there any linear operator $T:S'(\mathbb R^n)\to S'(\mathbb R^n)$ such that $T:B_{pq}^s(\mathbb R^n)\to B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but there exist a $F_{...
Liding Yao's user avatar
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On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
Liding Yao's user avatar
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I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy. A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ...
ash's user avatar
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2 answers
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I'm porting this question over from MSE as it did not get any responses other than one comment on there. Let $C^{k,\alpha}$ be a Hölder space where $0 \leq \alpha \leq 1$. I have seen various sources ...
CBBAM's user avatar
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Let us consider the space of polynomials $P^N$ of degree $\le N$. If $f\in P^N$ vanishes in $>N$ points, then $f\equiv 0$, but for any $N$ points, or fewer, there exists $f\neq 0$ vanishing at ...
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This question is mainly for understanding the history behind homogeneous spaces. There is extensive literature on Besov and Triebel-Lizorkin spaces. For instance, see the standard textbook: https://...
fast_and_fourier's user avatar
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I'm seeking help with a question regarding the space of bounded and uniformly continuous functions $C_u(T,d)$, where $(T,d)$ is a metric space. In this context, $C_u(T)$ is a closed subspace of $C_b(T)...
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The following startles me. Let $\Omega \subseteq \mathbb R^n$ and write $W^{\sigma,1}(\Omega)$ for the fractional Sobolev space with norm $$|u|_{W^{\sigma,1}(\Omega)} := \iint \frac{|u(x) - u(y)|}{|x-...
shuhalo's user avatar
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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 ...
Emily's user avatar
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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 $$ \int_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx \leq C \sup_{ x \in \...
shuhalo's user avatar
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Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$? If it helps, feel free to assume that $M$ is compact. (This is not a joke ...
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I would like to know whether for a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ (in the weak Lipschitz, so a "Lipschitz manifold", sense, not necessarily a Lipschitz graph domain), ...
Hannes's user avatar
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In $C_p$-Theory monographs, it is said that two topological spaces $X$ and $Y$ are said to be $t$-equivalent means that $C_p(X)$ is homeomorphic to $C_p(Y)$. Then they also define $u$-equivalences (...
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It is well-known that any Banach rearrangement-invariant function space $X$ on $[0,1]$ is a subset of $L_1[0,1]$, and I can find a reference that any quasi-Banach rearrangement-invariant function ...
user92646's user avatar
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For $p \in [1, \infty]$ the Hardy space $H_p$ is defined as the space of all analytic functions $f$ on the open disk satisfying $$\|f\|_{H_p} = \sup_{0 < r < 1} \|f(r\cdot)\|_{L_p(\mathbb{T})} &...
Seven9's user avatar
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1 answer
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$\newcommand{\loc}{\mathrm{loc}}$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 ...
AB_IM's user avatar
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1 vote
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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 ...
Ho Man-Ho's user avatar
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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 ...
SetValued_Michael's user avatar
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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 $L_\infty(0,1)$?...
user92646's user avatar
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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:...
Daniella Dannell's user avatar
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2 answers
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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 ...
epitaph's user avatar
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7 votes
0 answers
273 views

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 ...
Alan's user avatar
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0 answers
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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 $C(X)$.
Mir Aaliya's user avatar
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1 answer
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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.
Mir Aaliya's user avatar
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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) \|...
shuhalo's user avatar
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2 votes
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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 ...
mw19930312's user avatar
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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 "...
Chris Wendl's user avatar
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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} &...
shuhalo's user avatar
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1 vote
1 answer
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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, \...
A. Bailleul's user avatar
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6 votes
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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. ...
Glen M Wilson's user avatar
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2 votes
1 answer
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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 $C(\mathbb{R}^n;\mathbb{R}^d)$? $X$ ...
AB_IM's user avatar
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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 ...
Thomas Hank Clayton's user avatar
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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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