All Questions
67 questions
23
votes
9
answers
2k
views
Nonseparable counterexamples in analysis
When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...
19
votes
3
answers
1k
views
What standard Banach space is isomorphic to the completion of this different normed structure on $\ell^1$?
A colleague asked me the following question:
"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"
This ...
- 356
16
votes
3
answers
1k
views
A natural center of a convex weakly compact set in Banach space
Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...
- 161
13
votes
2
answers
653
views
The geometry of $\mathbb{R}^n$
Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces.
Then we define the set of equivalence classes
$$G(X,Y):=\left\{[T]; T,S \in ...
- 536
13
votes
0
answers
395
views
Converse to Riesz-Thorin Theorem
Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.
The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....
- 854
12
votes
1
answer
191
views
Spectra on different spaces
This is a method request: I am looking for techniques that allow me to investigate problems like this:
Let $T_1: \ell^1 \rightarrow \ell^1$ be a bounded operator with $\Re(\sigma(T_1)) \subset (-\...
- 305
12
votes
0
answers
435
views
Uniform closure of subspaces of Baire class 1
Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
- 1,704
10
votes
2
answers
666
views
Reference request: Extensions of Wiener's Tauberian Theorem
Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
- 710
10
votes
1
answer
439
views
Interpolation between $L_1^0$ and $L_2^0$
Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...
- 31.5k
10
votes
1
answer
899
views
Approximation of a compactly supported function by Gaussians
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
- 710
9
votes
1
answer
608
views
Interpolation theory and $C^k$-spaces
Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
- 779
7
votes
2
answers
419
views
A counterexample showing $BV_p \neq AC_p$
I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $p > 1$. ...
- 217
6
votes
3
answers
2k
views
Generalized Hardy-Littlewood-Sobolev Inequality
The Hardy-Littlewood-Sobolev Inequality says that
$$\text{for $p,q,r\in (1,+\infty)$ such that }\quad
1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$}
$$
$$
\exists C, \forall u\in L^p(\mathbb R^n),\...
- 16.2k
6
votes
2
answers
2k
views
How to prove the Hahn-Banach constructively
I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space.
Thanks in advance for any helpful answers.
- 71
5
votes
4
answers
362
views
Dual norm of a subspace of $\ell_\infty^3$
We define a norm on $\mathbb C^2$ as $\|(\alpha,\beta)\|:=\max\left\{|\alpha|,|\beta|,\big|\frac{\alpha+\beta}{\sqrt{2}}\big|\right\}.$ Can the dual norm be calculated explicitly?
- 1,879
5
votes
1
answer
216
views
Bounds on dimension of a subspace
Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that:
$$ \| u\|_{...
- 4,135
5
votes
1
answer
189
views
Subsequences of an orthonormal basis generating a strongly embedded subspace in $L_2(0,1)$
A closed subspace $M$ of $L_2(0,1)$ is said to be strongly embedded if the norms $\|\cdot\|_2$ and $\|\cdot\|_1$ are equivalent on $M$.
Let $(f_n)_{n\in \mathbb N}$ be a orthonormal basis of $L_2(...
- 4,461
5
votes
1
answer
669
views
Compact operators on $\ell^1$
Let $T$ be a compact symmetric operator on $\ell^2$ and $T\vert_{\ell^1}$ be bounded on $\ell^1$. Are there any non-trivial conditions that $T\vert_{\ell^1}$ is compact as well (for example would $T$ ...
- 329
4
votes
1
answer
670
views
A generalization of a theorem of Grothendieck
In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel _{p}$.
Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$.
Assume that $S$ is a subvector space of ...
- 356
4
votes
2
answers
340
views
Embeddings of Weighted Banach Spaces
Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces
$$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^...
- 2,044
4
votes
1
answer
548
views
Two definitions of $L^p$ spaces that are not always equivalent
There are two definitions of $L^p(S, \Sigma,\mu)$ in the literature. (Here $S$ is a set, $\Sigma$ is a $\sigma$-algebra of subsets of $S$ and $\mu$ is a positive measure.) The two definitions are ...
- 41
4
votes
1
answer
280
views
Approximation of an integral over the unit ball of L_1
For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and
$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- \frac{q(t)q((s-...
- 4,705
4
votes
1
answer
165
views
Dual spaces of Banach-valued $L^{p}$-spaces
Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\...
- 1,429
4
votes
1
answer
283
views
Absolutely continuity in variation of constant formula
We are talking here about the initial value problem on some Hilbert space $H$
$$y'(t)=Ay(t)+f(t), \\ y(0)=y_0 \in D(A).$$(Problem 1.13 in the reference)
Then $y(t)=e^{At}y_0 + \int_0^t e^{A(t-s)}f(s) ...
- 43
4
votes
0
answers
179
views
Condition on kernel convolution operator
I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with
$$
\Phi_i(2t)\approx \Phi_i(t), \quad i=1,2
$$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
- 101
4
votes
0
answers
117
views
Korovkin subset of $C(\mathbb{T})$
Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\...
- 521
4
votes
0
answers
211
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
3
votes
3
answers
2k
views
Determining if a set is a Basis for l^2
For each $ n\ge 1$ Define the vectors $e_n = (e_{nk})$ where $ k\ge 1$ and $ e_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,
- 4,135
3
votes
1
answer
640
views
Relationship between LlogL and Hardy spaces
I think that for positive, one-dimensional, periodic functions, the following statement is true:
$$
f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}),
$$
where
$$
LlogL=\{f\in L^1\,s.t.\,\...
- 843
3
votes
1
answer
657
views
Banach space of discontinuous functions(Killing continuous functions)
Edit: According to the comment of Prof. Majer, I revise the question:
For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$
$$\...
- 356
3
votes
1
answer
255
views
Closure of tensor product /tensor product semigroup
In this reference the following claim is made in Remark 2
Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
user127612
3
votes
2
answers
435
views
A possible norm on a subspace of $C^\infty([0,1])$?
I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...
- 3,388
3
votes
1
answer
151
views
The weakest condition guarantees some Separation-type of convex sets in Banach spaces
Classical Hahn-Banach Separation theorem plays a vital role in many branches of Analysis, Like functional Analysis, Convex Analysis, Variational Analyis, Theory of ODEs, optimal control and ...
- 369
2
votes
2
answers
176
views
Direct limit of the sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category of Banach spaces
Recently I have been reading the paper The categorical origins of Lebesgue integration by Tom Leinster (https://arxiv.org/pdf/2011.00412.pdf). In this paper, he said that：
For $n \geq 0$, let $E_{n}$ ...
- 123
2
votes
1
answer
154
views
Are these two norms on localized versions of $L^p_q$ equivalent?
$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$.
Let $E$ be the space of all real-valued ...
- 835
2
votes
1
answer
234
views
Counter example about blow-up solution of DEs
Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
- 395
2
votes
1
answer
140
views
An inequality about embedding of cube into metric spaces
A k-cube in $X$ is a function $\psi:\{-1,1\}^k\to (X,d)$.
An edge of a cube is a pair of points $\{\psi(\epsilon_1),\psi(\epsilon_2)\}$ in $X$ such that $\epsilon_1$ and $\epsilon_2 $ differ in ...
- 1,245
2
votes
1
answer
244
views
Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?
For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p_\omega$-norm of the sequence $v$ by
$$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right)...
2
votes
1
answer
143
views
How to characterize the order convergence in Bochner-integrable functions space?
Let $(\Omega,\Sigma,\mu)$ a finite measure space. We want to characterize the order convergence (for sequences) in Bochner integrable functions space $L^1(\mu,X)$, $X$ Banach lattice.
In $L^p$ we have:...
- 21
2
votes
1
answer
577
views
When is the bound in Riesz-Thorin Interpolation Theorem attained?
Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...
- 717
2
votes
1
answer
5k
views
Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$
Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...
- 1,127
2
votes
0
answers
88
views
Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
- 21
2
votes
0
answers
76
views
Fractional integration in Orlicz spaces
I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley.
And I would like to understand one question:
Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...
- 343
1
vote
1
answer
264
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
- 835
1
vote
2
answers
872
views
$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$
$c_0(C[0,1])$ is the $c_0$-direct sum of countably many $C[0,1]$.How to prove
$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$.
Here,Banach-space isomorphism means a bounded invertible operator ...
- 185
1
vote
1
answer
124
views
Do functions exist and are they dense? Or does it depend on the basis?
Consider an orthonormal basis $(\varphi_n)_{n \in \mathbb N}$ of $L^2(\mathbb R).$
We consider the functionals $\Phi_n$ given by $$ C^b(\mathbb R) \ni f \mapsto \left\langle \varphi_n, f \varphi_{n+1}...
- 25
1
vote
1
answer
237
views
Interpolation and embeddings for parabolic function spaces
I have a somewhat easy looking question on parabolic function spaces:
Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...
- 632
1
vote
1
answer
176
views
Some estimates on tensor norms
Denote $M_n$ to be $n\times n$ matrix. For $X\in M_n$ define $\|X\|_1:=\max\limits_{1\leq j\leq n}\sum_{i=1}^n|x_{ij}|$ and $\|B\|_2:=\max\{|\sum_{i,j=1}^nb_{ij}x_iy_j|:|x_i|=|y_j|=1,\ 1\leq i,j\leq n\...
- 1,879
1
vote
1
answer
114
views
Example of a nonconvex Chebyshev set in a metric space with continuous projection?
Question: Is there an example of a nonconvex Chebyshev set $S$ in a metric space $(X,d)$ whose projection map is continuous?
For convexity to be well-defined, we need to assume that $X$ is a vector ...
- 710
1
vote
1
answer
153
views
Optimal estimate in trace norm
Let $x,y$ be vectors of some Hilbert space of unit length.
Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$
Assume then that we know that $\left\lVert x-...
- 119