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.
148
questions
0
votes
0
answers
26
views
Questions regarding $C_n^1(\overline\Omega)$, the space of functions with normal derivatives
The definition of which functions have normal derivatives, and to which we can apply Green's First Identity to, seems to be very delicate. Let $\Omega$ be a $C^1$ domain in $\mathbb{R}^d$ with $d\geq ...
- 45
0
votes
0
answers
13
views
Calculate Components of square integrable functions w.r.t. some basis
Consider the space of square integrable functions on the non negative real numbers $L^2(\mathbb{R}_0^+)$. I found out, that the Laguerre functions modulo some normalization define an orthonormal basis ...
- 119
10
votes
1
answer
263
views
+100
On the map $\operatorname{Top}(X,Y \times Z) \longrightarrow \operatorname{Top}(X,Y) \times \operatorname{Top}(X,Z)$
Disclaimer: We define a topological space $X$ to be compact if every open cover has a finite subcover, but $X$ is otherwise allowed to be arbitrary.
I have been faced with the following problem:
Let $...
2
votes
1
answer
45
views
Hausdorffness of mapping space, implies Hausdorffness of codomain
I am trying to prove following:
Let Y be a topological space and $Top(X,Y)$ the space of all continuous functions from an arbitrary topological space $X\neq \emptyset$ into $Y$. If $Top(X,Y)$ equipped ...
1
vote
1
answer
27
views
Show that $\langle(f\circ\varphi_{\lambda})k_{\lambda}, (g\circ\varphi_{\lambda})k_{\lambda}\rangle=k_{\lambda}(\lambda)\langle f,g\rangle.$
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 ...
- 2,612
5
votes
2
answers
940
views
How is it possible for the L² norm of f − g to measure the area between the graphs of f and g?
Here is the definition of a norm given by my textbook;
(This is from Fourier Series and Boundary Value Problems by James Ward Brown and Ruel V. Churchill, Chapter 7)
I'm confused by what authors say ...
- 61
1
vote
0
answers
44
views
About the notation $C_{\to0}^0(\mathbb R^n)$
I remember loosely that I saw somewhere a space notation $C_{\to0}^0(\mathbb R^n)$ for the space of all continuous function that vanishes at infinity. But I can't find a reference. Does anyone know a ...
- 2,259
0
votes
0
answers
32
views
Are function spaces over a shrinking set vector bundles?
I have function spaces $F(S_t) \subseteq \{f : S_t \to \mathbb{R}\}$ defined over shrinking sets $S_t\subset S_s$ for $t> s$. I have a trivial fiber bundle $[0,\infty) \times F(S_0)$ where the ...
- 2,109
3
votes
0
answers
81
views
$C([0,\infty))$ is separable
In many books (e.g. Brownian Motion and Stochastic Calculus by Karatzas and Shreve) the fact that $C([0,\infty)):=\{f\colon [0,\infty)\to\mathbb{R} \mid f \text{ continuous}\}$ endowed with the metric
...
- 166
0
votes
0
answers
26
views
How Does the Limit lim. $\Phi_u(p)$ Indicate $u \in L^{\infty}(\Omega)$? [duplicate]
Question: Let $\Omega \subset \mathbb{R}^d$ be measurable with $|\Omega|<\infty$, and let $u: \Omega \rightarrow \mathbb{R}$ be measurable. Define
$$
\Phi_u:[1, \infty) \rightarrow[0, \infty], \...
0
votes
1
answer
46
views
Difference between $C^{0,a}(\Omega)$ and $C^{0,a}(\overline{\Omega})$
I'm reading a book and I discovered a strange (to me) statement:
(roughly stated): $\Omega$ is bounded open set in $\mathbb{R}^d$. Then
(i) If $u \in C^{0,a}(\Omega)$, then "statement 1" ...
0
votes
0
answers
63
views
Connection between $H^s_0(\Omega)$ defined by Fourriertransform and $H^s_0(\Omega)$ defined canonically.
I am studying Sobolev spaces and have a couple of question to those.
It is known that the space
$H^s(\mathbb{R}^n):=\lbrace f\in L^2(\mathbb{R}^n) : (1+|\xi|^2)^s |(\mathcal{F}f)(\xi)|^2 \in L^2(\...
- 69
1
vote
0
answers
75
views
Weak derivative of $|u|^s$ $(s>1)$
I often come across instances in texts where people calculate the weak derivative of $|u|^s$ for $s>1$ as $s|u|^{s-1} \operatorname{sign}(u) \partial_x u$ for some $u\in W^{1,s}(\Omega)$.
However, ...
Furkan
1
vote
1
answer
63
views
Continuous map $f:X\to M_n(\mathbb{R})$ as a map $f:X\times\mathbb{R}^n\to\mathbb{R}^n$ linear in second coordinate.
Let $S = \{f:X\times \mathbb R^n \to \mathbb R^n : f\ \text{continuous}, \ f(x,\_) \ \text{linear map}\}$, $T = \{g:X \to M_n(\mathbb R) : g\ \text{continuous}\}$ with $X$ a topological space, prove ...
0
votes
2
answers
56
views
How do I prove that the subset M = {f $\in$ C[0,1] : $\int_0^1f(x)dx = 0$} of C[0,1] is proximinal?
I want to show that the subset $$M = \{f \in C[0,1] : \int_0^1f(x)dx = 0\}$$ is proximinal in the Banach space C[0,1](equipped with the sup norm), that is, for every g $\in$ C[0,1] there exists f$\in$...
- 49