All Questions
Tagged with examples fa.functional-analysis
21
questions
6
votes
0
answers
170
views
Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?
I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...
- 55.5k
2
votes
2
answers
151
views
Hardy space inclusion in the right-half plane
I'm looking for an example of a function $u \in H_2$ such that $u \notin H_\infty$, where $H_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a ...
- 427
2
votes
1
answer
324
views
Are there "pathological convex sets" over ultravalued fields of char 2?
In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...
- 447
1
vote
1
answer
216
views
Examples of $C^{k,1}$ functions which are not $C^{k+1}$?
I'm currently reading this paper and the authors define the set $C^{k,1}(\mathbb{R}^n)$ as consisting of all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ having $k$ derivatives and for which:
$$
\|...
- 5,001
1
vote
0
answers
64
views
Multivarate "RKHS" Examples
I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^...
- 5,001
1
vote
1
answer
666
views
Known dense subset of Schwartz-like space and $C_c^{\infty}$?
After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
- 5,001
6
votes
0
answers
103
views
Real-world example of a Banach *-algebra with a nonzero *-radical
Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
- 3,222
1
vote
0
answers
53
views
Is it possible that a convex cone and its closure both induce vector lattices?
Given a convex cone $P\subset X$ where $X$ is a $K$-vector space, $K=\mathbb{R}\text{ or }\mathbb{C}$ is a field.
Suppose that $P$ satisfies positive element stipulations.
(1) $X=P-P$.
(2) $P\cap-P=...
- 7,891
10
votes
1
answer
429
views
Example of Banach spaces with non-unique uniform structures
While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform ...
- 7,891
1
vote
0
answers
189
views
system with solutions $\{x-a:0\leqslant a\leqslant z-1\}$ [closed]
What must be $F$ there where $0=F(1,x,0)=F(x-0,x,z)=F(x-1,x,z)=F(x-2,x,z)=F(x-3,x,z)=$ $\dots$ $=f(x-z-1,x,z)=0$?
Define $F$ in the domain where a continuous function exists that behaves so for $x\...
3
votes
2
answers
293
views
Examples of TVS with no non-trivial open convex subsets
I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
...
- 1,355
5
votes
1
answer
455
views
Measures which exhibit the "uncorrelated implies independent" property
Let $X$ be a topological linear space, and let $X^*$ be its dual space. Suppose that $X$ is complete and Hausdorff, and $X^*$ separates points. Let $Y$ be another such space, and let $f : X \to Y$ be ...
- 8,322
7
votes
7
answers
3k
views
Gelfand representation and functional calculus applications beyond Functional Analysis
I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on Gelfand representation and functional calculus in a crucial way.
I am curious about ...
30
votes
4
answers
4k
views
Elementary applications of Krein-Milman
This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try.
Recall that the Krein-...
4
votes
4
answers
1k
views
An example of a non-paracompact tvs (over the reals, say)
What is an example of a non-paracompact topological vector space?
I'm aware of this question, but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that ...
- 33.2k
31
votes
4
answers
11k
views
Counterexample for the Open Mapping Theorem
I would like to ask a counterexample for the classical theorem in functional analysis: the open mapping theorem in the case that $Y$ is Banach, but $X$ is not Banach to show that the completeness of X ...
- 311
2
votes
2
answers
1k
views
description of functions of conditionally negative type on a group
Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any ...
- 1,222
4
votes
3
answers
3k
views
Examples of Banach spaces and their duals
There are many representation theorems which state that the dual space of a Banach space $X$ has a particularly concrete form. For example, if $X = C([0,1],\mathbb R)$ is the space of real-valued ...
- 8,322
8
votes
2
answers
1k
views
Example for an integral, rectifiable varifold with unbounded first variation
I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.
Recapitulation
for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable ...
- 143
7
votes
3
answers
2k
views
What are some interesting sequences of functions for thinking about types of convergence?
I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
5
votes
2
answers
696
views
Can we distinguish the algebraic and continuous duals of a Banach space without choice (or HBT)?
The algebraic dual of a normed vector space is the space of all linear functionals to the ground field (either $\mathbb{R}$ or $\mathbb{C}$ for this question). The continuous dual is the subspace of ...
- 26.2k