All Questions
1,222 questions
2
votes
1
answer
252
views
type and cotype of spaces of continuous functions
I've recently read about the notion of (Rademacher) type and cotype of a Banach space in some article. In the books I checked afterwards, typical examples studied were $L^p$-spaces or the Schatten ...
- 105
1
vote
0
answers
85
views
A question on the Dieudonné property
Recall that a Banach space $X$ is said to have the Dieudonné property if for every Banach space $Y$, an operator $T:X\rightarrow Y$ that transforms weakly Cauchy sequences into weakly convergent ...
- 3,343
2
votes
1
answer
799
views
Weak-* convergence in $L^\infty((0,T)\times\Omega)$ implies weak-* convergence in $L^\infty(\Omega)$ for a.e. $t \in (0,T)$?
Let $\Omega$ be a bounded and smooth domain. Suppose I have a sequence of non-negative functions $u_n \in L^\infty((0,1)\times \Omega) \cap L^\infty((0,1);L^\infty(\Omega))$ with
$$0 \leq u_n \leq 1 \...
- 73
6
votes
1
answer
216
views
A minimality problem for a class of Banach spaces
The following question is related to the previous question Minimality properties of James' space; I post it as a new question since the system does not allow me to add a comment.
Question Consider ...
- 986
4
votes
1
answer
352
views
Minimality properties of James' space
I am interested in the following question about James' quasi-reflexive Banach space $\mathcal{J}$:
Does there exists a non-Hilbertian subspace $X$ of $\mathcal{J}$ such that $X$ isomorphically ...
- 1,005
1
vote
0
answers
136
views
Topology on function spaces for pointwise convergence
Suppose I have some collection of maps $T_\lambda: C^\infty(\Omega)\rightarrow C^\infty(\Omega)$ which are linear and parameterized by a parameter $\lambda >0$. (Perhaps more generally take $C^\...
- 11
4
votes
1
answer
574
views
Criterion for weak convergence of sequences
Let $E$ be a normed space and let $F\subset E^{*}$. It is known that $F$ is dense if and only if the restriction of $\sigma(E,F)$ on $B_E$ coincides with the weak topology.
Hence, if $F$ is dense and ...
- 5,529
5
votes
3
answers
675
views
$L^{\infty}$ as colimit
I recently read a result (in Jarchow's book) that any ultrabornological space can be expressed as a colimit (in the category LCS) of Banach spaces. My question is the following.
Let $\mu$ be a ...
- 5,405
0
votes
1
answer
126
views
Tauberian operators
Let $X$ be a Banach space non reflexive and $T$ from $l_2(X)$ to $l_2(X)$ a bounded operator defined by:
$$T(x_n )=\frac{x_n }{n}.$$
We know that :
$$T^{**-1}(l_2(X))=\{x_n^{**} \in l_2(X^{**}) : \...
0
votes
0
answers
147
views
Approximation of Inductive Tensor Product $C(X) \bar{\otimes} C(Y)$
The following question is from Banach Algebra Techniques in Operator Theory written by Ronald G. Douglas.
Assume both $X, Y$ are Banach spaces and $X \otimes Y$ is the algebraic tensor product. Let ${...
- 307
6
votes
1
answer
167
views
Extension Operator for $W^{1,\infty}(U,X)$
I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...
1
vote
0
answers
27
views
Approximation of multipliers by multipliers of a smaller set 2
This question is a refinement of my previous question.
Let $X$ be a compact metric space, and let $B$ be a bounded Banach Disk in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$...
- 5,529
1
vote
0
answers
99
views
Gluing together dense subset of Projective Limit in $Ban_1$
Let $(X_n,\pi_n^{m})$ be a countable projective system in the category Ban$_1$ of Banach spaces and short linear maps (is (continuous) linear constructions). Then (co)-completeness of this category ...
- 5,405
4
votes
1
answer
86
views
Approximation of multipliers by multipliers of a smaller set
Let $X$ be a compact metric space, and let $B$ be a convex balanced bounded set in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$.
Let $M=\{u\in C(X),~ uf\in B,~\forall f\in B\...
- 5,529
3
votes
1
answer
561
views
Construction of an infinitely Fréchet differentiable function with given set of zeros in a Banach space
After looking at this question, I am now wondering if the following is true.
Let $X$ be a separable Banach space over $\mathbb R$ or $\mathbb C$, and $A\subseteq X$ a closed set. Then there exists ...
- 1,755
4
votes
1
answer
472
views
Is the set of weak*-continuous operators closed in the weak*-operator topology?
I recently came across this unanswered MO question an answer to which I would also be interested in. However the formulation of said question is somewhat imprecise and lacking detail in my opinion so ...
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
4
votes
0
answers
2k
views
Eigenvalues and spectrum of the adjoint
In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$.
But in infinite dimensions this need no longer be ...
- 3,873
-1
votes
1
answer
119
views
Existence of a function with slow growth on derivatives
Does there exist a smooth compactly supported function $$f \in C^{\infty}_c((0,1))$$
such that
$$ \|D^k f\|_{L^{2}(0,1)} \leq \left\lfloor{\alpha\,k}\right \rfloor! \quad \forall\, k\in \mathbb N$$
...
- 4,143
4
votes
1
answer
364
views
Approximation property of a Banach space in terms of finite-rank projections
Let $X$ be a separable Banach space. Is this property equivalent to the approximation property?
There exists a chain $X_n$ of finite-dimensional subspaces of $X$, each being a range of some ...
- 97
-1
votes
1
answer
349
views
Sequence converging to different limits with respect to two different _complete_ norms
Do there exist a real vector space $X$ complete with respect to norms $|\cdot|$ and $\|\cdot\|$ and a sequence $(x_n)_{n\in \mathbb N} \subset X$ such that there exist $x,y\in X$: $x\ne y$, $|x_n - x|\...
- 1,277
9
votes
1
answer
977
views
Completed tensor product is exact
In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on ...
- 332
6
votes
2
answers
458
views
Subspaces of $\ell_p$ ($1<p<\infty$, $p\neq 2$) not isomorphic to $\ell_p$
Is it possible to show the existence of an infinite dimensional closed subspace of $\ell_p$ ($1<p<\infty$, $p\neq 2$), not isomorphic to $\ell_p$, in an elementary way?
For $1<p<q<2$,...
- 4,461
4
votes
1
answer
137
views
Are unit balls in Banach spaces retracts of bidual balls?
Let $X$ be a separable Banach space embedded canonically in $X^{**}$. Is there a retraction from the unit ball $B_{X^{**}}$ of $X^{**}$ onto the unit ball $B_X$ of $X$?
When we insist on uniformly ...
- 97
6
votes
0
answers
99
views
Is every separable Banach space with the MAP 1-complemented in a space with a monotone basis?
The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature.
Can every ...
- 11.3k
1
vote
1
answer
120
views
Does taking the modulus preserve weak $p$-summability of sequences in $L_q$?
For this question, all Banach spaces are over the reals.
Let $1\leq p<\infty$. Recall that a sequence $(x_n)$ in a Banach space $E$ is weakly $p$-summable if
$$ \Vert (x_n) \Vert_{p,w} := \sup_{\...
- 25.8k
5
votes
1
answer
699
views
Can $L^1_{loc}$ be represented as colimit?
Let $L^1_{loc}$ denote the set of all functions from $\mathbb{R}$ to itself which are locally integrable. For every infinite compact subset $K\subseteq \mathbb{R}$, let $L^1_{m_K}$ denote the space ...
- 5,405
0
votes
0
answers
170
views
Limit of balls in $L^p$
Setup:
Let $\mu$ be a measure on a measurable space $(X,\Sigma)$, such that for every $p ,q\in [1,\infty)$, $L^p_{\mu}(\Sigma)\subseteq L^q_{\mu}(\Sigma)$ if $p\geq q$. Furthermore, the inclusions ...
- 5,405
0
votes
0
answers
58
views
Bounds on $\inf_{x,x' \in \mathbb B_X}TV(P+x,Q+x')$, where $P$ and $Q$ are distributions with density on the space $X=(\mathbb R^n,\ell_p)$
Let $n \ge 1$ be an integer, $p \in [1,\infty]$, and $P$ and $Q$ be two (probability) measures on the metric space space $X=(\mathbb R^n,\ell_p)$ which have densities w.r.t the Lebesgue measure on $X$,...
- 6,853
5
votes
0
answers
139
views
Copies of $\ell_\infty^k$ in subspaces of the space of operators between $n$-dimensional Banach spaces
Are there a positive integer $k$ and an unbounded increasing function $d:\mathbb N\to\mathbb N$ (of growth order $\Omega(n^2)$) such that for any $n$-dimensional Banach spaces $X,Y$, the Banach space $...
- 4,535
10
votes
0
answers
230
views
Norm-attaining operators with values in a 2-dimensional Hilbert space
Is the set $N\!A(X,\ell_2^2)$ of norm-attaining operators from a Banach space $X$ onto the $2$-dimensional Hilbert space $\ell^2_2$ dense in the Banach space $L(X,\ell_2^2)$ of all linear continuous ...
- 4,535
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
1
vote
1
answer
194
views
Strictly increasing functions in reflexive subspaces of $C([0,1])$
By the Banach-Mazur theorem, every separable Banach space $X$ embeds into $C([0,1])$. When $X$ is reflexive, it is not possible to find a sequence of disjointly supported, non-negative functions in ...
- 97
6
votes
1
answer
618
views
Whether Krein-Milman property implies Radon-Nikodym property
A Banach space is said to have Krein-Milman property (KMP in short) if every closed bounded convex set of it is a closed convex hull of its extreme points. Eg. Any reflexive space has KMP, $\ell_1$ ...
- 521
4
votes
1
answer
291
views
Copies of $c_0$ in $C[0,1]$ and disjoint sequences
Let $M$ be a subspace of $C[0,1]$ isomorphic to $c_0$.
QUESTION: Is it possible to find a normalized disjoint sequence $(f_n)$ in $C[0,1]$ such that the distance of $f_n$ to $M$ tends to $0$ as $n$ ...
- 4,461
2
votes
2
answers
185
views
Show that $(S^1)^*=B(\ell^2)$ knowing $(\ell^1)^*=l^\infty$
Is there a way to show that dual of trace class operators, $S^1$, is $B(\ell^2)$, bounded operators on $\ell^2$, knowing that dual of $\ell^1$ is $\ell^\infty$?
- 21
1
vote
0
answers
86
views
Uniform continuity of sequence of semigroups
Let $T(t)$, $t\in [0,\tau]$, be a $C_0$ semigroup on an Banach space $X$. Also, let $T_n(t)$ be a sequence of semigroups that satisfies for all $x\in X$
$$\lim_{n\to \infty}\sup_{t\in [0,\tau]}\|T_n(...
- 395
3
votes
0
answers
133
views
Lower bound on the intersection of $\ell_1$ $n$-balls
Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ in $\ell_1$ norm, with distance $d$ and radius $R$.
Is there a lower bound on the volume of the intersection between the two n-balls? (assuming the ...
- 301
8
votes
0
answers
182
views
Distribution domination for sums of independent random variables in Banach spaces
Let $X$ be a Banach space and let $(\xi_n)$ and $(\eta_n)$ be independent mean-zero random variables with values in $X$ satisfying
$$
\sum_n \mathbb P(\xi_n \in A) \leq \sum_n \mathbb P(\eta_n \in A),
...
- 131
2
votes
0
answers
55
views
A holomorphic map into a Hilbert space with prescribed orthogonality
This is a variation of my previous question.
Let $X\subset \mathbb{C}^n$ be a domain, and let $L:X\times X\to \mathbb{C}$ be such that $L(x,x)>0$, $L(y,x)=\overline{L(x,y)}$ and $L(\cdot,y)$ is ...
- 5,529
4
votes
1
answer
174
views
A map into a Hilbert space with prescribed orthogonality
Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
Does there always ...
- 5,529
3
votes
1
answer
177
views
Quotient space of a locally uniformly rotund space
If $X$ is a uniformly rotund space , then for any closed subspace $M$ of $X$, $X/M$ is uniformly rotund. Does this hold for a locally uniformly rotund space? That is if $X$ is locally uniformly ...
- 585
5
votes
1
answer
177
views
An extremal property of points on the unit sphere of a 2-dimensional Banach space
Let $(X,\|\cdot\|)$ be a 2-dimensional real Banach space and $S=\{x\in X:\|x\|=1\}$ be its unit sphere. Assume that $S$ is smooth in the sense that for any $y\in S$ there exists a unique functional $y^...
- 41.9k
20
votes
2
answers
3k
views
Non-differentiable Lipschitz functions
As far as I understand, there are Lipschitz functions $f:\mathbb{R}\to\ell^\infty$ that are nowhere differentiable in the Frechet sense. Where can I find such an example?
- 28k
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
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
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
1
vote
1
answer
896
views
Known Lipschitz-free spaces
The Lipschitz-Free space (also known as Arens-Eells spaces) $\mathcal{F}(X,d)$ over a pointed metric space $(X,d)$ is a well-studied object. In many instances, we have "concrete" representations of ...
- 5,405
3
votes
0
answers
173
views
A Caratheodory-like result for infinite-dimensional simplices
Let $K$ be a compact metric space; $\Delta K$ be the set of Borel probability measures on $K$ endowed with the weak* topology; $X$ be a closed subset of $\Delta K$; and $x_0 \in \overline{\text{co}} X$...
- 537
1
vote
1
answer
303
views
Density of norm-attaining operators
By Bishop-Phelps theorem we know that for a real Banach space, the set of all norm attaining bounded linear functionals is norm-dense in $X^*$, the topological dual of $X$. We also know that in ...
- 585