All Questions
1,222 questions
81
votes
3
answers
9k
views
Norms of commutators
If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
- 31.5k
77
votes
0
answers
4k
views
2, 3, and 4 (a possible fixed point result ?)
The question below is related to the classical Browder-Goehde-Kirk fixed point theorem.
Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$
be a mapping such that
$$\Vert Tx-Ty\...
- 4,060
44
votes
1
answer
4k
views
Example of a compact set that isn't the spectrum of an operator
This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity:
Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty ...
- 5,743
43
votes
1
answer
5k
views
Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?
Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
- 559
34
votes
8
answers
9k
views
When is a Banach space a Hilbert space?
Let $\mathcal{X}$ be a real or complex Banach space.
It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds.
...
- 343
34
votes
1
answer
3k
views
tr(ab)=tr(ba), part 2.
This is a Banach space version of Andre Henriques' question
Trace Question
for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ ...
- 31.5k
32
votes
2
answers
4k
views
Are there non-reflexive vector spaces isomorphic to their bi-dual?
Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.
Are ...
- 7,902
31
votes
2
answers
3k
views
Is a normed space which is homeomorphic to a Banach space complete?
I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.
Does this imply that $(E,||\cdot||)$ is also a Banach space?
I think I read something ...
- 495
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
30
votes
3
answers
3k
views
Surjectivity of operators on $\ell^\infty$
Can anyone give me an example of an bounded and linear operator $T:\ell^\infty\to \ell^\infty$ (the space of bounded sequences with the usual sup-norm), such that T has dense range, but is not ...
- 301
30
votes
1
answer
1k
views
Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments
It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
- 6,853
29
votes
6
answers
9k
views
Nonseparable Hilbert spaces
Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
- 9,330
28
votes
3
answers
4k
views
A separable Banach space and a non-separable Banach space having the same dual space?
I asked myself the following question when I was student just for curiosity. I asked a bit around (my professor, some researchers that I know), but nobody was able to give me an answer. So maybe it is ...
- 6,034
28
votes
2
answers
1k
views
What is the Banach-Mazur distance between $\ell_\infty$ and $L_\infty$?
Given Banach spaces $X$ and $Y$, the Banach-Mazur distance between $X$ and $Y$ is defined as
$$ d(X,Y) = \inf\{ \|\varphi\|\|\varphi^{-1}\| : \varphi\colon X\to Y \text{ isomorphism} \}.
$$
We ...
- 3,497
24
votes
2
answers
2k
views
Unique predual of a Banach space
Suppose $E$ is a dual Banach space whose predual is unique, and $E_0$ is a codimension 1 weak* closed subspace of $E$. Is the predual of $E_0$ necessarily unique?
Okay, I will reveal the motivation. ...
- 42.8k
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 ...
22
votes
3
answers
7k
views
Subspace of $L^2$ that lies in $L^\infty$
Let $E$ be a closed subspace of $L^2[0,1]$. Suppose that $E\subset{}L^\infty[0,1]$. Is it true that $E$ is finite dimensional?
PS. This is actually a question from the real analysis qualifier. I came ...
21
votes
5
answers
4k
views
Isomorphisms of Banach Spaces
Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...
- 629
21
votes
3
answers
3k
views
Can you tell whether a space is Banach from the unit ball?
Let $V$ be a real vector space. It is well known that a subset $B\subset V$ is the unit ball for some norm on $V$ if and only if $B$ satisfies the following conditions:
$B$ is convex, i.e. if $v,w\...
- 8,493
21
votes
2
answers
1k
views
Meager subspaces of a Banach space and weak-* convergence
I previously asked a version of this question on Math.SE, but didn't receive an answer. (But there is a bounty there if you want to claim it!)
Let $X$ be a Banach space. (If it helps, feel free to ...
- 29.8k
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
19
votes
6
answers
8k
views
Unbounded operator bounded in a dense subset
Let $X, Y$ be normed vector spaces, where $X$ is infinite dimensional. Does there exist a linear map $T : X \rightarrow Y$ and a subset $D$ of $X$ such that $D$ is dense in $X$, $T$ is bounded in $D$ (...
- 783
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
19
votes
1
answer
3k
views
Infinite convex combinations in a Banach space
Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds:
For any sequence $(x_k)_{k\ge0}$ in $C$, and for
any sequence of non-negative real numbers $(\...
- 60.5k
18
votes
3
answers
2k
views
What are the right categories of finite-dimensional Banach spaces?
This is inspired partly by this question, especially Tom Leinster's answer.
Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...
- 11.4k
18
votes
1
answer
564
views
Is the space of Hankel operators complemented in B(H)?
Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$.
Let ...
- 25.8k
17
votes
1
answer
2k
views
Are "most" operators on an infinite-dimensional complex Banach space "diagonalizable"?
This is true for finite-dimensional spaces: the diagonal operators on a finite dimensional complex vector space form contain a dense open set and the nondiagonalizable operators have measure 0.
To be ...
- 63.9k
17
votes
4
answers
2k
views
Banach-Mazur applied to a Hilbert space
The Banach-Mazur theorem says that every separable Banach space is isometric to a subspace of $C^0([0;1],R)$, the space of continuous real valued functions on the interval $[0;1]$, with the sup norm.
...
- 6,924
17
votes
1
answer
912
views
$(1+\epsilon)$-injective Banach spaces, complex scalars
It is well known that a real Banach space which is $(1+\epsilon)$-injective for every $\epsilon >0$ is already 1-injective (Lindenstrauss Memoirs AMS, 1964, download here). Using common ...
- 1,704
17
votes
0
answers
488
views
Large almost equilateral sets in finite-dimensional Banach spaces
Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set $\{x_i\...
- 4,878
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
16
votes
6
answers
2k
views
Finding closed subspaces whose sum isn't closed
Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum ...
- 42.8k
16
votes
2
answers
682
views
Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?
Let $X$ be a Banach space. Consider the map
$$
\alpha\colon X\hat{\otimes} X^* \to B(X)^*,
$$
defined one simple tensors as
$$
\alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, a\...
- 3,497
16
votes
2
answers
4k
views
Usefulness of Frechet versus Gateaux differentiability or something in between.
If you have a function $V: L \rightarrow \mathbb{R}$, where $L$ is an infinite dimensional topological vector space, there are multiple notions of differentiability. For $x,u \in L$, $V$ is Gateaux ...
- 943
16
votes
1
answer
2k
views
What (classes of) Banach spaces are known to have Schauder basis?
Motivation:
I am trying to see for what class of Banach spaces the following result is true:
There exists an increasing sequence of finite dimensional subspace {$V_n$} of a Banach space X (with ...
- 365
16
votes
0
answers
542
views
$C^*$-algebra generated by those operators that are bounded on every $\ell_p$
Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...
- 25.8k
16
votes
0
answers
1k
views
Finite Rank Commutators
My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...
- 31.5k
15
votes
5
answers
2k
views
Between Tietze's and Dugundji's extension theorems
The celebrated Tietze extension theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen ...
- 60.5k
15
votes
3
answers
8k
views
What is an isomorphism of Banach spaces?
The nLab page on Banach spaces (http://ncatlab.org/nlab/show/Banach%20space) was recently criticised as being, in effect, too heavily biased to category theory (not of the Baire kind) and not enough ...
- 26.8k
15
votes
5
answers
680
views
Idiosyncratic characterizations of $\ell^p$, for $p\not=1,2,\infty$
Do there exist, either in the literature or in folklore, theorems
that characterize some particular $\ell^p$ space(s) ($p\not=1,2,\infty$)?
Such a theorem should reveal the particular space(s) as ...
- 17.6k
15
votes
2
answers
2k
views
In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?
This is a cross-posted on MSE here.
Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
- 1,225
15
votes
1
answer
1k
views
Intersection of complemented subspaces of a Banach space
The following seems a very basic question in the theory of complemented subspaces of Banach spaces, but I was not able to find a reference, so I wish to ask it here.
Question. Let $X$ be a Banach ...
- 60.5k
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
15
votes
1
answer
2k
views
Quotients of $\ell_\infty$ by separable subspaces
Given a (closed) separable subspace $M$ of $\ell_\infty$, I am interested in conditions implying that the quotient $\ell_\infty/M$ is isomorphic to a subspace of $\ell_\infty$.
It is not difficult ...
- 4,461
15
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 8,512
15
votes
1
answer
441
views
Weak*-closure of finite rank operators on dual space
Given a Banach space $X$, we consider the space $B(X^*)$ of bounded, linear operators on $X^*$ with the weak*-topology from its canonical predual $B(X^*)_*=X^*\hat{\otimes}X$. What is $\overline{F(X^*)...
- 3,497
15
votes
1
answer
889
views
Operator norms of circulant matrices
The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix.
For complex numbers $a_1,\ldots,a_n$, I will use the notation
$$
\mbox{...
- 717
15
votes
1
answer
4k
views
Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?
The Open Mapping Theorem, the Bounded Inverse Theorem, and the Closed Graph Theorem are equivalent theorems in that any can be easily obtained from any other. The Closed Graph Theorem also easily ...
- 2,049
15
votes
2
answers
931
views
Distinguishing topologically weak topologies of Banach spaces
Are the weak topologies of $\ell_1$ and $L_1$ homeomorphic?
Strangely may it sound, the question seeks contrasts between norm and weak topologies of Banach spaces from the non-linear point of view. ...
- 11.3k
14
votes
4
answers
550
views
About the existence of characters on $B(X)$
Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$?
I know the proof of the fact that $M_n(\mathbb{C})$ ...
- 355