All Questions
1,222 questions
2
votes
1
answer
119
views
Finite-representability of $\ell_p$ in super-reflexive spaces
Let $E$ be a Banach space. Is it possible that $E$ is super-reflexive and $\ell_p$ is crudely finitely representable in $E$ for all $p\in (1,2)$?
It seems unlikely but I cannot find an argument off ...
- 331
1
vote
1
answer
220
views
Criterion of reflexivity
Let $E$ be a Banach space.
It is known that if for any equivalent norm on $E^*$ the closed unit ball of $E^*$ is weakly* closed, then $E$ is reflexive (a very short proof is in the book by Fabian, ...
- 5,529
5
votes
1
answer
358
views
Pisier's property $(\alpha)$
Let $\Omega$ be a probability space. Suppose $(\epsilon_i)_{1\leq i\leq n}$ is a sequence of i.i.d. Bernoulli random variables on $\Omega,$ i.e. $(\epsilon_i)_{1\leq i\leq n}$ are independent and $P(\...
- 455
14
votes
0
answers
205
views
Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?
A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article:
W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$.
In Convex ...
- 25.8k
2
votes
2
answers
374
views
A criterion for norming sets
Let $F$ be a Banach space with the closed unit ball $B$. Let $E\subset F^*$ be a total subspace such that $B$ is complete with respect to the norm $|||f|||=\sup \limits_{e\in E,~e\ne 0} \frac{|\left&...
- 5,529
4
votes
0
answers
144
views
Embedding of $\ell_2$ in $L^p([0,1])$
Let $(g_n)_{n\geq 1}$ be a sequence of i.i.d. complex Gaussian random variables on $[0,1].$ Then it is easy to see that the map $j:\ell_2\to L^p([0,1])$ defined as $je_n=[E(g_n^p)]^{\frac{1}{p}}g_n,n\...
- 455
4
votes
1
answer
157
views
Norm of "tensoring" with the identity
Consider a Banach space $E$ and a discrete set $X$. For an operator $T$ on $\ell^2(X)$ I can consider and induced operator $T'$ on the Bochner-Lebesgue space $\ell^2(X;E)$ of $E$-valued square-...
- 165
6
votes
1
answer
240
views
The approximation property for some spaces of holomorphic functions
I am reading a circle of papers which use arguments based on Fredholm determinants of nuclear operators to compute numerical quantities associated to real-analytic and holomorphic dynamical systems. ...
- 6,206
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
6
votes
2
answers
735
views
Tensor product space with projective norm is incomplete
Ryan says in his book "Introduction to Tensor Products of Banach Spaces"(pg. 17) that for Banach spaces $X$ and $Y$, $X\otimes Y$ equipped with projective norm is not complete unless $X$ and $Y$ are ...
- 163
3
votes
1
answer
177
views
Rate of convergence of weakly null sequences
If $x_n$ is a normalized, weakly-null sequence in a Banach space, and $\epsilon_n\to 0$, does there exists a non-zero functional $f$ such that $|f(x_n)|<\epsilon_n$ for all $n$?
- 1,361
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
6
votes
1
answer
323
views
Hartogs' theorem in Banach spaces
In complex analysis one learns Hartogs' theorem:
Let $U\subseteq \mathbb{C}^n$ open and $f: U \rightarrow \mathbb{C}$ a function. Then $f$ is analytic iff for all $1\leq i \leq n$
$$ z \mapsto f(...
- 1,045
0
votes
1
answer
233
views
Is $(\ell^1(\mathbb N_0),\sigma(\ell^1,\ell^\infty))$ not quasi-complete?
In Jarchow's Locally Convex Spaces this not being quasi-complete is asserted on page 206 referring to Corollary 11.4.4 on page 228 saying that a Banach space is reflexive if and only if its closed ...
- 3,584
5
votes
1
answer
456
views
The Bochner integral about a semigroup of bounded linear operators on a Banach space
Let $T(t)$ be a semigroup of bounded linear operators on a Banach space $X$. When does the following hold
$$
\int_0^t T(s)x ds = \Big(\int_0^t T(s) ds\Big)x, \quad x \in X \, ,
$$
where $ t \in (0,1)$?...
- 51
2
votes
1
answer
233
views
complemented $\ell_p$ subspaces in $\ell_p$ sums of spaces
Note: By "subspace" I always mean an infinite-dimensional closed subspace.
Notation.
Let us write
$$\oplus_p\ell_q^n:=\left(\bigoplus_{n=1}^\infty\ell_q^n\right)_{\ell_p}\;\;\;\text{ and }\;\;\;\...
- 1,591
5
votes
0
answers
150
views
On the relation between Lipschitz free-spaces
Let $X$ be a pointed metric space, with base point 0. The space of Lipschitz function which preserves the base point,
$Lip_0(X)=\{f:X\to\mathbb{R} : f(0)=0\}$ consider with the norm $\|f(x)\|=\sup_{x\...
- 71
3
votes
0
answers
125
views
Commutative discrete cyclic operator groups on topological vector spaces
Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
- 1,959
5
votes
2
answers
216
views
On the coincidence (or non-coincidence) of two norms defined on the quotient of a given Hilbert $ C^{\ast} $-module by a certain linear subspace
Let $ A $ be a $ C^{\ast} $-algebra, $ I $ a closed two-sided ideal of $ A $, and $ \mathcal{E} $ a Hilbert $ A $-module. Let
$$
\mathcal{E}_{I}
\stackrel{\text{df}}{=}
\{ x \in \mathcal{E} \mid \...
- 1,396
3
votes
1
answer
151
views
The weakest condition guarantees some Separation-type of convex sets in Banach spaces
Classical Hahn-Banach Separation theorem plays a vital role in many branches of Analysis, Like functional Analysis, Convex Analysis, Variational Analyis, Theory of ODEs, optimal control and ...
- 369
2
votes
1
answer
323
views
Characterization of nets with no convergent subnets in Banach spaces
Let $X$ be a finite-dimensional Banach space and $(x_i)_{i\in I}$ a net in $X$. Since every limited net in $X$ has a convergent subnet, it follows that $(x_i)_{i\in I}$ does not admits a convergent ...
- 563
11
votes
1
answer
441
views
Example of Banach spaces with non-unique uniform structures
While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform ...
- 8,071
4
votes
0
answers
92
views
Simultaneous representations of elements of projective tensor products
Let $E,F$ be Banach spaces and consider the projective tensor product $E \widehat\otimes F$. If $\tau \in E \widehat\otimes F$ with $\|\tau\|<1$ then by definition we can find $(x_n)\subseteq E$ ...
- 18.7k
0
votes
0
answers
97
views
Is there any concise sufficient condition for the dual space to have Kadec property?
A normed space $E$ has a
Kadec property if the norm- and weak topologies coincide on the unit sphere of $E$.
Kadec-Klee property if any sequence on the unit sphere, that is weakly convergent is also ...
- 5,529
1
vote
1
answer
974
views
Annihilators and pre-annihilators
I asked this question on Math StackExchange first, but it was not answered.
If $X$ is a Banach space and $Z$ is a subset of $X^*$, consider the annihilator of $Z$ in $X^{**}$:
$$
Z^{\perp}=\{x^{**}...
- 1,361
2
votes
0
answers
115
views
Mean value of a map into Banach space
Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Let $(Y,\|\cdot\|)$ be a Banach space. Given a Bochner integrable map $f:X\to Y$ with $\|f\| \in L^2(X,\mu)$. The mean value of $f$ over $X$, ...
- 169
1
vote
1
answer
214
views
A generalization of strict convexity
Consider the following properties of a Banach space:
the intersection of any support hyperplane with the unit sphere is
(S) a singleton (this is the strict convexity);
(SF) finite-dimensional set;...
- 5,529
1
vote
1
answer
158
views
Extending functionals on $X^*$
Suppose $X$ is a non-reflexive Banach space, $Z$ a closed subspace of $X^*$, and $f$ a bounded functional on $Z$ with the property that there exists non-zero $x\in X$ such that $f(z^*)=z^*(x)$ for all ...
- 1,361
2
votes
0
answers
143
views
About a property of bounded closed convex set
Terminology:
For a bounded closed convex (bcc for short) set $A$, define $w(A)$ to be the infimum of the distance between pairs of
parallel hyperplanes supporting $A$.
We say that a bcc ...
2
votes
1
answer
347
views
On complemented copy of $c_{0}$ in projective tensor products
Suppose that the projective tensor product of $l_{\infty}$ and $X$ contains a complemented copy of $c_{0}$. Does it follow that $X$ contains a complemented copy of $c_{0}$?
- 81
8
votes
1
answer
314
views
What algebras are quotients of $\ell_1(\mathbf{N})$?
Every separable Banach space is a linear quotient of $\ell_1$, however not every separable Banach algebra is a Banach-algebra quotient of $\ell_1(G)$ for some group $G$ (these are the so called ...
- 331
3
votes
0
answers
169
views
A spanning set for an annihilator set on a Banach space
Let $(z_n)$ be a $H^\infty$-interpolating sequence on the open complex unit disc $\mathbb D$. If $A$ is some Banach space of analytic functions on the disc, denote by $X$ the closed subspace of all ...
- 31
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,
- 4,143
9
votes
0
answers
261
views
SVD-type decomposition for the tensor product of three Hilbert spaces?
(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the ...
- 25.8k
6
votes
3
answers
3k
views
Non-empty resolvent set, then operator closed?
On Hilbert spaces, the following is true:
Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed.
The obvious proof I see to show this uses explicitly the ...
- 115
1
vote
1
answer
190
views
Bounded operators on the Stinespring representation space
Let $A$ be a $C^*$-algebra and let $\phi:A\to B(H)$ be a completely positive map. The Stinespring representation theorem constructs a representation of $A$ on a Hilbert space $K$, which is constructed ...
- 99
3
votes
1
answer
222
views
Intuition for inequality involving permutation and Hamming Cube
Let $C^n=\{0,1\}^n$ be a metric space (Hamming Cube). The distance on $C^n$ is defined by
$$
d(\varepsilon,\varepsilon'):=|\{j:\varepsilon_j\ne\varepsilon'_j\}|,
$$
$\varepsilon=(\varepsilon_1,\...
- 1,245
1
vote
0
answers
864
views
A guide to the work of Timothy Gowers on Banach Spaces [closed]
I'm undergraduate student and I'm thinking of doing my graduation thesis on some of Prof. Gowers work on Banach Spaces. It is not required to produce an original result in my thesis, I'm only asked to ...
- 329
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
4
votes
1
answer
412
views
Abstract Definition of a Reproducing Kernel Hilbert Space
This is a very basic question about the definition of a reproducing kernel Hilbert space (RKHS).
It seems the standard definition of a RKHS is as a Hilbert space $H$ of functions on some set $X$ ...
- 1,307
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
0
votes
0
answers
263
views
Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 305
1
vote
1
answer
71
views
Every open convex-valued multimap has global sections?
Let $X$ be a compact Polish space and $Y$ be a separable real Banach space. Assume $U \subseteq X \times Y$ is open, bounded in $Y$-norm, and s.t. for any $x \in X$, $\{y \in Y \mid (x,y) \in U\}$ is ...
- 1,368
0
votes
2
answers
108
views
Density of positive orbits of $C_0$-semigroup
Suppose that $(T(t))_{t\geq0}$ is a $C_0$-semigroup on a Banach space $X$ and assume that there exists $x\in X$ such that $\{T(t)x:\ t\geq0\}$ is dense in $X$. I wonder why the set $\{T(t)x:\ t\geq ...
1
vote
0
answers
220
views
About the projection on the unit sphere
Let $H$ be a Hilbert Space and let $A\subset H$ be a connected set such that any two elements of $A$ are linearly independent and also $A^{\bot}=\left\{0\right\}$ (this seems to be immaterial). Is ...
- 5,529
4
votes
2
answers
543
views
Gaussian measure on Banach space
Assume we have a Gaussian measure $\mu$ supported on a Banach space $X$. Can we always find a Hilbert space $H$ embedded in $X$ sch that $\mu$ is also supported on $H$?
- 43
10
votes
0
answers
207
views
Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.8k
1
vote
0
answers
109
views
Two tensor product norms inducing different topologies on the space of simple tensors
Are there two Normed spaces $V,W$ for which the algebraic tensor product $V\otimes W$ admits two different norms, both satisfying $\parallel x \otimes y \parallel= \parallel x \parallel. \...
- 356
1
vote
1
answer
109
views
Continuous factors for invertible simple tensors
Our following question is motivated by this very interesting answer
Assume that $A$ is a $C^{*}$ algebra. Put $X=\{a\otimes b \mid a,b \in G(A)\}$ where $G(A)$ is the space of all ...
- 356
3
votes
1
answer
204
views
Two questions on $l_{p}$-saturated Banach spaces
Let $1<p<\infty$. Recall that a Banach space $X$ is $l_{p}$-saturated if every infinite-dimensional subspace of $X$ contains a subspace isomorphic to $l_{p}$. I have a seemingly stronger notion ...
- 3,343