All Questions
1,222 questions
10
votes
1
answer
900
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
3
votes
2
answers
253
views
Reference request: $\alpha$-Hölder spaces as double duals
If $(X,d)$ is a complete metric space, we define the $\alpha$-Hölder class $\Lambda_\alpha(X)$ as the subset of $C_b(X)$ satisfying that
$$
\sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}.
$$
...
- 1,090
3
votes
1
answer
354
views
Peak sets and Choquet boundary of a function algebra
I have two problems to ask.
Let $A$ be a function algebra of $C(K)$. $t\in K$ is said to be a peak point of $A$ if $\exists~f\in A$ s.t. $|f(t)|=\|f\|$ and $|f(s)|<|f(t)|$ for any $s\neq t$. ...
- 521
1
vote
0
answers
56
views
Monotonicity of the norms on the sequence spaces 2
This is a complement of my previous question about the sequence spaces (I'm afraid, there will be a third part).
Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties:
$...
- 5,529
0
votes
1
answer
80
views
Vectors concentrated on one coordinate
Suppose $X$ is a Banach space, $(e_i)$ a normalized basis, $(e_i^*)$ the biorthogonal functionals, and $Y$ a finite codimensional subspace of $X$. Given $N$ and $\varepsilon$, can we find $x\in Y$ ...
- 247
2
votes
1
answer
378
views
Does the norm on a sequence space have to be monotone?
Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties:
$\rho(\lambda u)=\lambda\rho( u)$, for every $u\in [0,+\infty)^{\mathbb{N}}$ and $\lambda\ge 0$;
$\rho(u+v)\le \...
- 5,529
2
votes
1
answer
397
views
Can a bijection between function spaces be continuous if the space's domains are different?
It is well-known that any bijection $\mathbb{R} \rightarrow \mathbb{R}^2$ cannot be continuous. But suppose we have the two spaces $A = \{f(x):\mathbb{R^2}\rightarrow \mathbb{R} \}$ and $B = \{f(x):\...
- 545
0
votes
1
answer
314
views
Regarding orthogonality in Banach space
Let $(X,\|.\|)$ be a Banach space over the real line. Let $x\neq 0$ and $y\neq 0$ be in $X$, then $x$ is said to be orthogonal to $y$ if $\|x+\lambda y\|\geq\|x\|$ for every real number $\lambda$.
...
- 195
1
vote
1
answer
315
views
Characterizing a norm on sequences
Let $\{a_i\}$ be a sequence of reals such that $|a_i|\geq|a_{i+1}|$ for all $i$, and consider the following norm: $$\|\{a_i\}\| = \sup_k \frac{1}{\sqrt{k}}\sum_{i=1}^k |a_i|~.$$ One can see that -- ...
- 4,049
6
votes
0
answers
113
views
Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...
- 395
0
votes
0
answers
115
views
If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...
- 623
10
votes
0
answers
266
views
Are biduals of spaces of differentiable functions on hypercubes Grothendieck?
Consider the space $E_n = C^1([0,1]^n)$ of continuously differentiable functions with the usual norm
$$\max\{ \|f\|_\infty, \|f^\prime_{x_1}\|_\infty, \ldots, \|f^\prime_{x_n}\|_\infty\}.$$ making it ...
- 11.3k
0
votes
1
answer
275
views
Does this norm have a specific name? Banach space? References?
Let $(X,\mathscr{B},\mu)$ be a $\sigma$-finite measure space. Let $\gamma$ be a probability measure on $L_2(\mu)$ with $\mathrm{supp} \, \gamma = L_2(\mu)$ and existing first moment. Then
$$
f \...
- 123
7
votes
1
answer
308
views
Complemented subspaces constructed from finite pieces- part II
This is a follow up to: Complemented subspace constructed from finite pieces
Suppose $Y=\overline{\cup E_n}$ is a closed subspace of a separable Banach space X, where each $E_n$ is a $n$-dimensional ...
- 247
1
vote
1
answer
121
views
Complemented subspace constructed from finite pieces
Suppose $Y=\overline{\cup E_n}$ is a closed subspace of a Banach space, where each $E_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E_n\subseteq E_{n+1}$. Can one ...
- 247
5
votes
1
answer
855
views
$L\log L$ and Hardy space on the upper half plane
Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane.
It is well-known that the Cauchy ...
- 51
4
votes
2
answers
449
views
Weak closure of subsets of the unitary sphere of a Banach space
Assume that $(X,\|\cdot\|)$ is a Banach space with $\|\cdot\|$ strictly convex. Define $S=\{x\in X:\|x\|=1\}$. Suppose that $\varepsilon>0$ and $x_0\in S$ and define
$$
B_\varepsilon=\{x\in X:\|x-...
- 409
1
vote
0
answers
94
views
Interpolation theory: equivalence of norms
Consider the interpolation space $Z=(X,Y)_{\theta,p}$. In the case $Y\subseteq X$ do we have that, for all $a>0$ the following norm:
$$N_a:x\mapsto\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \...
4
votes
2
answers
1k
views
If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?
Let $X$ be a Banach space.
By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.
I am ...
- 623
1
vote
0
answers
110
views
On functions obtained from Gaussian Quadrature integration
Fix an integer $n \ge 2$. Let $x_1,...,x_n$ s and $w_1,...,w_n$ s be the Gauss Quadrature nodes and weights respectively in the interval $[0,1]$ (https://en.wikipedia.org/wiki/Gaussian_quadrature) . ...
- 1,209
2
votes
1
answer
246
views
"Compactness in measure" in function spaces
In Chapter 4.9 of the book "Measures of noncompactness and condensing operators" (Vol. 55 of *Operator theory: advances and applications), the authors mention the property "compactness ...
- 163
3
votes
3
answers
489
views
Sum of subspaces is closed iff inclination is positive
It is a well-known result in functional analysis that the sum $M+N$ of two subspaces of a Banach space with $M\cap N=0$ is closed if and only if the inclination
$$\widehat{(M,N)} := \inf_{x\in M, \|x\|...
- 799
1
vote
1
answer
176
views
Reference on vector-valued convex conjugate
The following definition of convex conjugate is taken from Wiki:
Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$
Denote the dual pairing by
$$\langle \cdot ,...
- 623
5
votes
1
answer
232
views
Interpolation of some Lebesgue spaces
When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
- 52.3k
9
votes
1
answer
481
views
Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?
The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...
- 327
2
votes
0
answers
326
views
Dual of the space of affine functions
Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\...
- 521
4
votes
2
answers
244
views
Compact images of nowhere dense closed convex sets in a Hilbert space
Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$.
Question. Is there a non-compact linear bounded operator ...
- 41.9k
10
votes
3
answers
739
views
Is there a version of Fischer-Riesz theorem for Banach space?
$( \Omega,F, P )$: a measurable space equipped with a finite measure
$(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra
$p$ : a constant bigger than $1$
...
- 406
3
votes
1
answer
162
views
Is it true that every Banach space has at least one extreme point that is normed by some point?
Definition: Let $X$ be a Banach space and $X^*$ be its continuous dual of $X,$ that is, $X^*$ contains all bounded linear functionals on $X.$
Denote
$$B_{X^*} = \{x^*\in X^*: \|x^*\|_{X^*}\leq 1\}.$$...
- 623
11
votes
1
answer
227
views
Complemented subspaces of $C(\beta\mathbb N\times \beta\mathbb N)$
Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\...
- 4,535
8
votes
0
answers
167
views
A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters
Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$?
(...
- 4,535
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 $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
- 395
15
votes
2
answers
660
views
Multiple of identity plus compact
Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of ...
- 247
6
votes
1
answer
381
views
Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?
Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$
is not ...
- 395
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
14
votes
1
answer
694
views
Criterion for a Banach algebra to be finite dimensional
Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional.
Question. Does it follow that $A$ is finite dimensional?
...
- 12.6k
0
votes
2
answers
1k
views
Does point-wise weak convergence give weak convergence in $L^2(I;X)$?
Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...
- 395
1
vote
1
answer
112
views
Orthogonal complement vector space
Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study
$X^{\perp_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle_{L^2} =0 \ \forall x \in X \right\}$
and
$X^{\perp_{H^{-...
- 13
5
votes
0
answers
231
views
Which subspaces of $\ell_p^n$ are isometric?
This question is similar to the one asked here:
Extending linear isometries from subspaces of $\ell_p^n$
Let $p$ be an even integer. Let $X,Y$ be subspaces of $\ell_p^n$, and let $U : X \to Y$ be a ...
- 53
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
1
vote
0
answers
74
views
Nonlinear maps in Riesz Thorin theorem
The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...
user127450
8
votes
2
answers
601
views
If $E\oplus_\phi E \cong E\oplus_\psi E,$ does it imply that $\phi= \psi$?
Let $E\neq \{0\}$ be a Banach space.
For each $p\in[1,\infty), $ we define
$$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$
Let $F$ be another Banach space.
By $E\...
- 623
0
votes
1
answer
115
views
Does there exists an extreme point $(a_1^*,...,a_n^*)$ of $B_{\mu^*}$ such that $a_i^*\neq 0$ for all $1\leq i\leq n$ and $\sum_{I=1}^n a_i^*a_i=1?$
Fix a natural number $n\geq 1.$
Let $\mu$ be a norm on $\mathbb{R}^n$ satisfying
$$\mu(0,...,0,\stackrel{i}{1},0,...,0) = 1 \quad\text{for all }1\leq i\leq n.$$
Let
$$B_{\mu} = \{(a_1,...,a_n)\in \...
- 623
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 $$ C^b(\mathbb R) \ni f \mapsto \left\langle \varphi_n, f \varphi_{n+1}...
- 25
6
votes
1
answer
270
views
Approximation property counterexamples? (Also: relation to tensor products)
I remember reading somewhere (but unfortunately, I've forgotten where it was) that the canonical map from the (completed) projective tensor product of two Banach spaces to the (completed) injective ...
- 508
2
votes
1
answer
170
views
A formula for vector valued measurable functions
Let $B_{\infty}(\Omega)$ be the space of bounded measurable functions on the measurable space $\Omega$. For a given Banach space $X$, let us denote $B_{\infty}(\Omega,X)$ by the set of all bounded ...
- 4,058
1
vote
1
answer
124
views
Compactness of operators and norming sets
Originally asked on MSE.
Let $T$ be a linear map from a normed space $E$ into a Banach space $F$.
Let $D\subset \overline{B}_{F^{\ast}}$ be norming, i.e., there is $r>0$ such that $\sup\limits_{v\...
- 5,529
1
vote
1
answer
52
views
Infinitely many independent functions that are only frequency localized?
A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
- 11
3
votes
1
answer
221
views
Does Bishop-Phelps Theorem hold for extreme points (slightly different version)?
Recall the Bishop-Phelps Theorem.
Bishop-Phelps Theorem: Let $B\subseteq E$ be a bounded, closed, convex subset of a real Banach space $E.$
Then the set
$$\{e^*\in E^*: e^* \text{ attains its ...
- 623
3
votes
3
answers
358
views
Preannihilators of subspaces of separable duals
If $Y\subset X^*$ is a closed subspace (where $X$ is a separable Banach space), the preannihilator of $Y$ in $X$ is $Y_{\perp}:=\{x\in X : y^*(x)=0, \forall y^*\in Y \}$. If $Y$ is a proper subspace ...
- 1,361