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.

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3
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0answers
74 views

$\sum_{n \in \mathbb Z} |\hat {f}(n)|^p |n|^{p-2}) \leq C_p \|f\|^p_L$ where $p \in (1,2],$ $f \in L^p(\Bbb T)$

I'm trying to prove that for p $\in$ (1,2] and for some $C_p$ > 0, $$\sum_{n \in \mathbb Z} |\hat {f}(n)|^p |n|^{p-2} \leq C_p \|f\|^p_{L^p} \text{ for all } f \in L^p(\Bbb T)$$ Now we first note ...
3
votes
1answer
52 views

The homotopy type of $\mathrm{Maps}\left[X,Y\right]$ depends only on the homotopy types of $X,Y$

Let $X,Y$ be two topological spaces and $\mathrm{Maps}\left[X,Y\right]$ be the collection of continuous maps $X\to Y$ with the compact-open topology. Is the homotopy type of $\mathrm{Maps}\left[X,Y\...
2
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0answers
81 views

How to prove that $\mathscr{H}$ is dense in $C_0$

Let us define $C_0$ as the space of all $f \in C^0(\mathbb{R}^n,\mathbb{R})$ such that $\lim_{\|x\| \to +\infty} \|f(x)\|=0$. Let be $\sigma>0$ fixed and define $\mathscr{H}$ the vector space ...
1
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0answers
27 views

A question about $L^1$ functions.

I have a tiny questions that has been bothering me, but I haven't found a pleasing answer yet. The question is: If you have a function $g\in L^1(a,b)$ for some $a,b\in\mathbb{R}$ and some function $...
0
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2answers
25 views

Linear Functional is Discontinuous

I'm trying to show that the linear functional $f:(\ell^1, \|\cdot\|_\infty) \longrightarrow (\mathbb{R}, \|\cdot\|)$ given by $$f((x_n)_{n \in \mathbb{N}})=\sum\limits_{n=1}^\infty x_n$$ is ...
1
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1answer
77 views

Boundedness in Function Spaces

Let $(C(I),d_\infty)$ be a metric space, where $C(I)$ is the set of all continuous functions on a closed interval $I$ and $d_\infty=\sup_{x\in I}\{|f(x)-g(x)|\}$. I am reading a book about real ...
1
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1answer
20 views

Continuous functions from compact Hausdorff spaces to the interval

Let $X$ be a compact Hausdorff space and let $C(X,I)$ be the set of all continuous functions from $X$ into the closed interval $[0,1]$. If we equip $C(X,I)$ with the the topology of uniform ...
3
votes
1answer
50 views

PDE problem on $L^2$ convergence

With $I \subset \Bbb{R}$, let $\chi_I$ denote the indicator function of I, $$\chi_I(x)=\begin{cases} 1 & \text{if x $\in$ I} \\ 0 & \text{otherwise} \end{cases}$$ For any $k \in \Bbb{N}$, ...
0
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1answer
19 views

Rigor behind choice of index for Cauchy squence in bounded function space.

We have a Cauchy sequence in the normed space of bounded functions $(f_n)_n \subset B(\Omega, \mathbb{K})$. I have shown that $(f_n(\omega))_n$ is a Cauchy squence for all $\omega \in \Omega$. What's ...
1
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1answer
39 views

Infinite-dimensional change-of-basis and Laplace transformations

I apologize for the vagueness of the question, but I'm working more from high-level intuition here than from rigorous formalism. In short, my question is this: although the Laplace "basis" is not ...
0
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0answers
13 views

Homomorphis of the algebra of measurable fucntions

Let $(X,\Omega,\mu)$ be a measure space and $L_0(X)$ be the algebra of measurable functions. Suppose $T\colon L_0(X)\to L_0(X)$ injective homomorphism that is linear and $T(fg)=T(f)T(g)$ for all $f,g\...
0
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0answers
33 views

Munkres section 46 exercise 8: Imbedding of $C(X,Y)$ into $\mathscr{H}$

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, give $X\times Y$ the square metric $d=max\{d_X,d_Y\}$, and let $\mathscr{H}$ denote the nonempty closed, bounded subsets of $X\times Y$. The metric $d$ ...
0
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1answer
55 views

Proving the function set as a vector space

For $x\ge y$, is the set of functions that satisfy the $\operatorname{f}(x)\gt \operatorname{f}(y)$ condition a vector space? How should we approach this question before applying vector space axioms ...
2
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1answer
133 views

Uniform convergence sufficient conditions.

Let $f_n$ be a sequence of continuous functions defined in a compact interval C, so that $f_n$ converges to $f$ pointwise. Show that $f_n$ is uniformly convergent to $f$ if and only if: 1) $f$ is ...
0
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1answer
41 views

Compact Subset in $C[0,1]$

Let $S=\{f\in C[0,1]:\max\limits_{x\in[0,1]}(|f|+|f'|)\leq M\}$. Show that $S$ is compact in $C[0,1]$. I have proved that any sequence in $S$ contains a convergent subsequence by Arzela Ascoli ...
0
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0answers
16 views

Holomorphic function spaces on Stein manifolds

This may be a long shot, but I'm interested in learning about holomorphic function spaces on Stein manifolds, however, I can't find much literature on the topic. I don't know if this is just too hard ...
2
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0answers
20 views

What means a Riemannian manifold be in a function space?

I am trying understand the arguments for short time existence for curvature flows in the book "Curvature Problems" by Claus Gerhardt and there he assumes that $M$ is a compact Riemannian manifold and ...
0
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1answer
16 views

Compact Operator Improves Regularity in Function Spaces

Let $\Lambda \subset \mathbb{R}$ be a bounded domain, and let $T:L^2(\Lambda)\to L^2(\Lambda)$ be a compact operator. Can we say anything about the regularity of $Tf$, $f\in L^2(\Lambda)$? Does a ...
0
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1answer
27 views

Dense subset of $L^2(\Omega)$

Consider the function space $A =L^2(\Omega)$, where $\Omega\subseteq \mathbb{R}^n$ is open. We have seen that the space of test functions, $B=C_0^{\infty}(\Omega)$ is dense in $A$. Now my question is, ...
2
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1answer
31 views

Help with exercise from Munkre's of general version of Arzela's Theorem

So i have been trying to do an exercise from Munkre's that goes like this: Let $X$ be an Hausdorff space that is $\sigma$-compact; Let $f_n$ be a sequence of functions $f_n : X \rightarrow \mathbb{R}^...
2
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0answers
43 views

Help understanding Reproducing Kernel Hilbert spaces?

I am trying to wrap my head around some concepts of Reproducing Kernel Hilbert Spaces (RKHS) without having a formal background in functional analysis. Since I am trying to form an intuition about ...
0
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1answer
51 views

Size of a function space [closed]

Im interested in measuring the size or how big a function space is. Is there any concrete measure for such case?
0
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0answers
28 views

How to interpret domain of a function where the input is scalar to the power vector.

From the paper that introduced PointNet(Qi et al,2016). We have: $f(\{x_1,...,x_n\}) \approx g(h(x_1),...,h(x_n)),$ where $f: 2^{\mathbb{R}^N} \rightarrow \mathbb{R}^N$, and $h: \mathbb{R}^N \...
-1
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2answers
42 views

For 2 functions $f$ and $g$ in $E$, let $d(f,g) = \max\limits_{x \in [a,b]}|f(x)-g(x)|$. Prove that $(E,d)$ is a metric space.

Let $E$ be the set of all real-valued continuous functions on $[a,b]$ and for 2 functions $f$ and $g$ in $E$, let $d(f,g) = \displaystyle\max_{x \in [a,b]}|f(x)-g(x)|$. Prove that $(E,d)$ is a metric ...
0
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0answers
26 views

Trigonometric polynomials and separation of points

I am trying to understand why the algebra of real trigonometric polynomials separates points on $[0, 2\pi)$ (in order to apply Stone-Weierstrass Theorem). Does anyone have a proof for this? I can ...
1
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0answers
23 views

Difference quotient for Hölder continuous functions

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in C^\alpha_{\mathrm{loc}}(\Omega)$. For $h>0$, $1\leq k\leq n$, let $$D_k^hu(x)=\frac{u(x+he_k)-u(x)}{h}$$ where $e_k$ is the $k$-th ...
2
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0answers
22 views

On “orthogonal families” in $C[0,1]$

The Legendre polynomials can be used to construct an infinite set $X \subset C[0,1]$ with the property that $$\int_{0}^{1} f(x)g(x) \ dx = 0$$ for any distinct $f,g \in X$. I want to see if this can ...
1
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0answers
21 views

Does the Hellinger distance generalize to a new class of function space norms?

Perhaps the most commonly used norm on function space is the $L^p$ norm. Put in terms of distance between functions, you would say that $$D(f,g) = \sqrt[p]{\int |f(x)-g(x)|^p \,\mathrm{d}x}.$$ The ...
5
votes
1answer
191 views

Is possible to show that the linear operator $T(\varphi)(x) = \int_{V_x\cap M} \varphi(y)\text{d}y$ has spectral radius $>0$.

Fix some $σ>2/(3\sqrt{3})$, let $M$ be the interval $[x_-,x_+]$, where $$x_- = \text{unique real root of $x^3 + \sigma = x$}$$ and $$x_+ = \text{unique real root of $x^3 - \sigma = x$}.$$ ...
0
votes
1answer
71 views

How do you find the spanning set of real-valued functions with only it's finite domain?

I am trying to understand how to find a spanning set of all real-valued functions such that this set is defined on a finite domain. I know how to do this with polynomials defined on ℝ, but I can't ...
5
votes
2answers
270 views

Prove that $C_0(X)$ is separable given that X is locally compact metric space

I'm struggling to prove the following fact: Suppose that $X$ is locally compact metric space. Let us denote with $C_0(X)$ the space of functions vanishing at infinity (i.e., $\forall f \in C_0(X)$ $\...
3
votes
1answer
67 views

Proof of $f\chi_{\Lambda_n} \to f\chi_{\Lambda}$ in admissible spaces for compact sets $\Lambda_n$

I have a question regarding a paper I found dealing with the problem of Phase Retrieval and I would like to know how one statement could be proven. Here is a link to the paper: https://arxiv.org/abs/...
0
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0answers
39 views

Reproducing Kernel Hilbert Subspace of $L^2$

Let $\mu$ be a finite Borel measure on a non-empty Borel subset $X\subseteq (0,\infty)$. The space $H\subseteq C(X;\mathbb{R})$ such that every element of $H$ can be identified with an element of $ ...
1
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1answer
41 views

Prove: $(\forall \epsilon > 0 )(\forall x \in K)(\exists U \in O(x))(\forall f \in Z)(\forall y \in U) | f(x) - f(y) | < \epsilon$ .

Let $K$ be compact space and $Z \subseteq C(K, \mathbb{R})$ compact, in metric topology $T_{d_{\infty}}$ on $C(K, \mathbb{R})$ which is induced by metrics $d_{\infty}$. Prove: $(\forall \epsilon > ...
1
vote
1answer
19 views

Follow up: Density Characterized by Weak Topology

This question is a follow-up on: this question. Let $F$ be a non-empty subset of $C(X,Y)$, where $X,Y$ are Hausdorff (and for simplicity assume that $Y$ is metric). Let $\tau$ be the weak topology ...
1
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1answer
38 views

Density in Compact-Open Equivalent to containment of subbases

Let $X,Y$ be Hausdorff spaces and consider the compact-open topology on $C(X,Y)$. Let $F\subseteq C(X,Y)$ be a non-empty subset and consider the topology $\tau$ on $C(X,Y)$ with subbase $$ B_F\...
0
votes
1answer
91 views

Function from a set to a ring

Let $A$ be a ring and $S$ be a set. Let $F(S,A)$ be the set of functions of the form $f: S \rightarrow A$. For all $f,g \in F(S,A)$ define $(f+g)(s)=f(s)+g(s)$, and $(fg)(s)=f(s)g(s)$. Show that $F(S,...