All Questions
1,222 questions
11
votes
2
answers
1k
views
Do non-stable Banach spaces exist?
Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties:
Is every infinite ...
- 7,583
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
$$
B_\varepsilon=\{x\in X:\|x-...
- 409
0
votes
0
answers
106
views
A noncontinous algebra map between Banach algebras
What is an example of two Banach algebras $A$ and $B$, and an algebra map $\phi:A \to B$ which is not continuous?
- 203
1
vote
1
answer
234
views
Density and the projective tensor product
Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with Schauder basis $\{b_i\}_{i =1}^{\infty}$. Is the set
$$
D^+\...
- 5,405
11
votes
3
answers
4k
views
Structure of an intersection of $L^p$-spaces
In what follows, $L^p$ denotes the space of functions from $\mathbb{R}$ to $\mathbb{R}$ such that $\int_{\mathbb{R}} |f(x)|^p\mathrm{d}x < \infty$.
I am interested to understand the structure we ...
- 2,306
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
3
votes
2
answers
1k
views
Extensions of Urysohn's inequality
A version of Urysohn's inequality states that for a symmetric convex body $K \subset \mathbb{R}^n$, one has
$$
\left(\frac{\text{vol}(K)}{\text{vol}(B_2)} \right)^{1/n} \le \frac{1}{\sqrt{n}} E
\; \| ...
- 1,731
2
votes
1
answer
188
views
Dual fixed point
Let $E$ be a Banach space, let $T:E\to E$ have norm $1$ and let $\nu\in E^*\setminus\{0\}$ be such that $T^*\nu=\nu$. Under which conditions there is $e\in E$ such that $Te=e$ and $\langle e,\nu\...
- 5,529
2
votes
0
answers
137
views
Conditions on the inequality with a gauge norm
Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
- 343
5
votes
1
answer
1k
views
Reference request: The resolvent is analytic in the resolvent set
I am busy reading through Taylor's paper Spectral Theory of Closed Distributive Operators.
On page 192, he defines the resolvent and spectrum of $T$:
Later on in the paragraph, he then proceeds by ...
- 289
10
votes
1
answer
901
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
2
votes
0
answers
520
views
Example of a non-reflexive Banach space and two sequences
Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$.
If $X$ is reflexive, ...
- 199
3
votes
1
answer
727
views
Question about pointwise convergence of operators
Consider two Banach spaces $E,F$ and a net $T_\alpha : E \to F$ of continuous operators.
I know that for each $x \in E$ the net $T_\alpha (x)$ is convergent in $F$ and it is easy to show that the ...
- 2,071
31
votes
0
answers
2k
views
Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?
Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...
- 4,878
0
votes
1
answer
407
views
Criteria for $\epsilon$-Density
Let $Y$ be a compact, separable metric space and $X=C(Y)$ Banach space. There are many criteria for a linear subspace $Z\subseteq X$ to be dense; notably the Stone-Weierstraß theorem.
Are there ...
- 5,405
5
votes
2
answers
437
views
Sets in constructive mathematics
It is not completely clear how Bridges, Richman and Youchuan treated sets in their paper. Example is in the following lemma (Lemma 7 on p. 7):
Let $U$ and $V$ be (inhabited to mean $\exists u \in U, \...
- 791
4
votes
2
answers
187
views
Largest ideal in bounded linear maps on Schatten-$p$ class
Let $1\leq p<\infty.$ Denote $S_p(\ell_2)$ be the set of all compact operator $x$ on $\ell_2$ such that $Tr(|x|^p)<\infty.$ Define $\|x\|_{S_p(\ell_2)}:=Tr(|x|^p)^{\frac{1}{p}}.$ This makes $S_p(...
- 1,879
2
votes
1
answer
291
views
Čech complex of rigid $K$-space - Closed image of boundary maps
Let $(X,\mathcal{O}_X)$ be a rigid $K$-space with a finite affinoid covering $(U_i)_{i\in I}$ such that any intersection of the $U_i$ is affinoid too.
Equipping the direct products with the maximum ...
- 473
4
votes
1
answer
265
views
Does Banach-Mazur distance between regular polygons admit any structure that lends to approximation or exact results in particular situations?
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides. Do closed form or approximate results exist (at least at special infinitely ...
- 1,826
5
votes
2
answers
2k
views
Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \...
- 371
0
votes
1
answer
114
views
$ \overline{(A-A)}\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $?
Let $X$ be a separable Banach space and $A$ is a subset of $X$ such that
$$
A\cap\overline{B}(0,r) \text{ is weakly compact, } \forall r>0.
$$
Can we say that :
$$
\overline{(A-A)}\cap\overline{...
- 117
-1
votes
1
answer
320
views
Existence of weak limit for bouded sequence $\{y_n\}$ such that for every weak limit point $\{y_n\}$ must equal $y$
Let $X$ be separable Banach space and $\{x_n\}$ be a bounded sequence, relatively weakly compact sequence in $X$. we set $y_n=\frac{1}{n}\sum_{i=1}^{n}{x_i}$, then (together with the Krein and ...
- 199
7
votes
1
answer
393
views
On norming weakly$^*$ sequences in the dual of the Banach space $c_0$
A bounded subset $B$ of the dual $X^*$ of a Banach space $X$ is called norming if the formula $\|x\|:=\sup\{|x^*(x)|:x^*\in B\}$ determines an equivalent norm on $X$.
Observe that the sequence $(e_n^*...
- 41.9k
0
votes
2
answers
344
views
subspace topology and strong topology
Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? If this is not correct, then on ...
- 71
6
votes
3
answers
2k
views
reflexive banach space
I want to ask this non-expert question:
What does it mean geometrically for a Banach space to be reflexive?
Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...
- 351
7
votes
0
answers
248
views
Isometries on the unit sphere
Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
- 1,361
5
votes
1
answer
265
views
Complemented subspaces of Lorentz sequence spaces?
Let $d(\textbf{w},p)$, $1\leq p<\infty$, denote the Lorentz sequence space, where $\textbf{w}=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1$ is a normalized decreasing weight.
Is there very much known ...
- 1,591
1
vote
1
answer
203
views
Continuous function on colimit
Let $X$ be a Banach space and $f:X\rightarrow \mathbb{R}$ be continuous. Suppose that $\{X_n\}_{n \in \mathbb{N}}$ is a strictly nested sequence of sub-Banach spaces, for which $\cup_{n \in \mathbb{N}...
- 5,405
6
votes
1
answer
203
views
How to calculate the volume of a section of a convex body?
The following is essentially a partial case for my previous question.
Let $B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $\mathbb{R}^m$, say $l^p$-norm, $p\in (1,\infty)$....
- 5,529
2
votes
0
answers
62
views
Decomposition of the Orlicz norm into sequential norm
I am bearing seeking for a sequential decomposition of the norm in Orlicz space.
Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$.
Given $u\in L^p(\Bbb R^d)$ let
$$n\...
- 1,101
9
votes
1
answer
462
views
Uniqueness up to isometric isomorphism of predual of $(\sum_{\lambda\in\Lambda} H_\lambda)_{l_\infty}$ where $H_\lambda$ are Hilbert spaces
This fact is an easy consequence of results of the paper Classes of Banach spaces with unique isometric preduals. by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a ...
- 1,697
3
votes
1
answer
162
views
Lipschitz choice of "norm attaining maps'
I am not very sure if the following problem has been treated in the literature and if so, whether it always holds:
A Banach space $X$ is isomorphic to a Hilbert space if the 'norm attaining' map $F$...
- 101
3
votes
2
answers
253
views
Reference request: $\alpha$-Hölder spaces as double duals
If $(X,d)$ is a complete metric space, we define the $\alpha$-Hölder class $\Lambda_\alpha(X)$ as the subset of $C_b(X)$ satisfying that
$$
\sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}.
$$
...
- 1,090
8
votes
1
answer
305
views
Subspaces isomorphic with quotients
Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?
- 1,361
6
votes
3
answers
2k
views
Space of compact operators
I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach ...
- 11.3k
2
votes
0
answers
57
views
Is this Beppo-Levi curl space a Banach space?
Let us define the quotient space:
$$ V = \{ \mathbf{u} \in L^2_{loc}(\mathbb{R}^3; \mathbb{R}^3) : \operatorname{curl} \mathbf u \in L^2(\mathbb{R}^3; \mathbb{R}^3) \} / \nabla H^1_{loc}(\mathbb{R}^3)....
- 163
5
votes
1
answer
456
views
The Bochner integral about a semigroup of bounded linear operators on a Banach space
Let $T(t)$ be a semigroup of bounded linear operators on a Banach space $X$. When does the following hold
$$
\int_0^t T(s)x ds = \Big(\int_0^t T(s) ds\Big)x, \quad x \in X \, ,
$$
where $ t \in (0,1)$?...
- 51
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
5
votes
1
answer
224
views
Conditional expectation of random vectors
$\newcommand{\E}{\mathsf{E}}$
$\newcommand{\P}{\mathsf{P}}$
The following additional question was asked in a comment by user Oleg:
Suppose that $(\Omega,\mathcal F,\P)$ is a probability space, $B$ ...
- 128k
4
votes
2
answers
434
views
A homeomorphism between the unit interval $[0,1]$ and a linearly independent subset of a Hilbert space
Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$
Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ ...
- 433
5
votes
1
answer
1k
views
Space of compact operators defined on separable Hilbert space
Let $X$ be a separable Banach space and consider $\mathcal{K}(X)$ the space of compact operators $K\colon X \rightarrow X$. Is it true that the space $\mathcal{K}(X)$ is separable? If yes, why? If no, ...
- 51
4
votes
1
answer
3k
views
Besov and Triebel-Lizorkin spaces
Let me start with a couple of notational reminders. For $\xi\in \mathbb R^n$,
$$
1=\varphi_{0}(\xi)+\sum_{\nu \ge 1}\varphi_{\nu}(\xi),\quad \varphi_{0}\in C^\infty_c(\mathbb R^{n}),\quad \varphi_{\nu}...
- 16.2k
1
vote
0
answers
115
views
Algorithm/iterative procedure for constructing hypercyclic vectors?
Let $B$ be a separable Banach space and let $L:B\rightarrow B$ be a hypercyclic operator; here I use the definition of hypercyclicity given implicitly by Birkhoff's Transitivity Theorem: continuous ...
- 5,405
6
votes
0
answers
158
views
Quotients of subspaces of $C(\alpha)$
A well known problem, attributed to H. P. Rosenthal, asks whether or not every quotient of $C(\alpha)$, $\alpha$ countable ordinal, is $c_0$-saturated. As it is known, $C(\alpha)$ are $c_0$-saturated ...
- 986
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
0
votes
1
answer
81
views
If $\tau_1\subset \tau_2$ and $X^*$ is separable for $\tau_1$ then $X^*$ is separable for $\tau_2$?
Let $X$ be a Banach space the associated dual space is denoted by $X^*$. Take $\tau_1$ and $\tau_2$ two topologies in $X^*$ compatible with the duality $(X^*,X)$, such that $\tau_1\subset \tau_2$.
...
- 199
2
votes
0
answers
159
views
Explicit homeomorphism between $L^p$ and Sobolev Space
From the Anderson-Kadec theorem, we know that all separable infinite-dimensional Banach spaces are homeomorphic. I'm wondering, is there an explicit such homeomorphism between $W^{p,k}(\mathbb{R}^n)$ ...
- 5,405
2
votes
2
answers
341
views
Weaky compact subset of Banach space with separable predual
Let $X$ be a Banach space and $S\subseteq X$ be a subspace such that the unit sphere of $S$ is weakly compact. If $Y^*=X$ for some separable Banach space $Y,$ is it true that $S$ is separable?
- 1,879
7
votes
1
answer
291
views
Does separability of the strong operator topology imply separability of the underlying space?
Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.
Suppose that the strong operator topology on $B(X)$ is separable and that the cardinal number of $B(X)$ is continuum.
...
- 4,058
0
votes
0
answers
154
views
Use of this space of very rapidly decreasing continuous functions
Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space
$$
V_p
:=
\left\{
f \in C([0,\infty)):\,
\sum_{n=1}^{\infty} ...
- 5,405