# 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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### $\sum_{n \in \mathbb Z} |\hat {f}(n)|^p |n|^{p-2}) \leq C_p \|f\|^p_L$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $k \in \Bbb{N}$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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/...
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### 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 ...
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\...
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,...