Questions tagged [function-spaces]
Questions about spaces of functions, such as continuous functions between topological spaces or certain reproducing kernel Hilbert spaces. Does not concern equivalent classes of functions such as $L^p$ spaces.
166 questions
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Dot notation for multivariable functions [closed]
Imagine I have a function $f(x,y)$. If I want to define the function keeping $y$ constant and say that it belongs to a specific space over $x$, is it correct to write, for example,
$$
f(x,\cdot)\in W^{...
- 3,585
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Continuity of evaluation map at a single point under the compact-open topology
Without assuming anything on the topology of $X$ and $Y$, equip $C(X; Y)$ the compact-open topology.
Then for every fixed $x\in X$, the evaluation map at $x$, $Ev_x: f\mapsto f(x)$ from $C(X; Y)$ to $...
- 2,982
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Reference Request: Besov spaces are compactly embedded in Hölder spaces
In the paper Wasserstein GANs are Minimax Optimal Distribution Estimators [1], the authors state within Lemma 3.3, p.12, that Besov spaces [2] of generalized smoothness (i.e., with the added parameter ...
- 322
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Does it suffice to take a subbasis of $Y$ in the definition of the compact-open topology?
Let $X$ and $Y$ be topological spaces. The compact-open topology on $\operatorname{Hom}(X, Y)$ is defined as the topology having the following subbasis:
$$
V_{K, U} = \{f: X \to Y \mid f(K) \subseteq ...
- 4,970
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What is the support of a Gaussian Process's probability distribution, and what does a ``sample from a GP'' mean?
Suppose $D$ is a subset of $\mathbb{R}^n$, $F = \{f:D \to \mathbb{R}\}$, and $k(x,y)$ is a kernel function that is continuous, positive definite, symmetric, and $k(x,x) > 0$ for all $x$. Then ...
- 388
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Is there an H-group such that the induced group structure ([X,Y],+) is not natural? [closed]
I was reading the book "Homotopical Topolgy" from A. Fomenko and D. Fuchs and in the first chapter it has the followind definition:
Definition 1: Suppose that for every space $X$ (resp. $...
- 319
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Does any nice finite-dimensional subspace of $L^2$ have an orthonormal basis of nonnegative functions?
Context: the question came up while discussing function approximations in the context of computer graphics. In particular, (truncated) spherical harmonics (SH) are a popular way to encode a spherical ...
- 17k
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Is the compact-open topology the coarsest topology on $\mathcal C(X,Y)$ that is admissible on compacta?
Let me begin the question with some definitions and background.
Let $X$ and $Y$ be topological spaces and let's consider the set of all (not necesarily continuous) maps $Y^X$ from $X$ to $Y$. Let's ...
- 525
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Norm and scalar product on $C(\mathbb{R})$?
Consider $C(\mathbb{R})$, i.e. continuous functions from $\mathbb{R}$ to $\mathbb{R}$. This is a vector space with the usual pointwise operations. Are there
norms, and/or
scalar products
on this ...
- 2,350
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Example of a function space $Y^X$ where $Y$ is normal but $Y^X$ is not with the compact-open topology
Given a set $X$ and a topological space $Y$ the set of all (not necesarilly continuous) maps from $X$ to $Y$ is denoted by $Y^X$.
As a brief introduction to the problem, I first want to specify what I ...
- 525
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A question about weighted Lebesgue spaces
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 ...
- 8,158
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An example of an admissible topology on compacta that doesn't coincide with the compact-open topology
Given two topological spaces $X$ and $Y$, the set $Y^X=\{f:X\rightarrow Y : f \textrm{ is a function} \}$ is the set of all (not necesarily continuous) functions from $X$ to $Y$. The subsets $U^K=\{f\...
- 525
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Proof The Riesz-Fischer Theorem In Stein and Shakarchi
I'm currently going through Real Analysis by Stein and Shakarchi, specifically his proof of the Riesz-Fischer on page 70 of the text.
The specific part of the proof that is confusing me is as follows. ...
- 2,934
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The space of derivatives
Suppose $I$ is an open interval in $\mathbb R$, and suppose $f:I \longrightarrow {\mathbb R}$ is differentiable. Then $f'$ doesn't need to be continuous, a standard example being $f(x)=x^2\sin(1/x)$ ...
- 425
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Boundedness of complex factorization function in inverse Laplace transformation context
stackexchange intelligence ;-),
I am concerned with an old research paper from 1955 about density and closedness of certain polynomials. For his argumentation, the author creates a complex function $F(...
