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53
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2answers
4k 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 ...
45
votes
0answers
2k 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 ...
27
votes
1answer
2k 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 $ab$ ...
26
votes
0answers
2k 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 ...
24
votes
2answers
1k 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 ...
20
votes
3answers
1k 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 ...
20
votes
2answers
1k views

surjectivity of operators on l^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 ...
19
votes
7answers
4k views

When is a Banach space a Hilbert space?

Let $\mathcal{X}$ be a 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. Question: Are there ...
19
votes
2answers
972 views

What theorem constructs an initial object for this category? (Formerly “Integrability by abstract nonsense”)

Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than ...
17
votes
5answers
2k 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 ...
16
votes
5answers
2k views

Brownian motion and spheres

Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$: $$ ...
15
votes
4answers
909 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. ...
15
votes
0answers
714 views

Decomposable Banach Spaces

An infinite dimensional Banach space $X$ is decomposable provided $X$ is the direct sum of two closed infinite dimensional subspaces; equivalently, if there is a bounded linear idempotent operator on ...
15
votes
0answers
525 views

Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces

Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm. Question: Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so ...
14
votes
5answers
547 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 ...
14
votes
3answers
2k 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 ...
14
votes
1answer
560 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 ...
14
votes
1answer
409 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 ...
14
votes
0answers
443 views

Dual of the Ultraproduct of a Banach Space

Suppose we have a Banach space ultraproduct $(E_i)_U$. I can think of three natural objects which are "dual-like": $(E_i^*)_U$, the ultraproduct of the duals of the ground spaces. The space made up ...
13
votes
3answers
1k 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 ...
13
votes
2answers
438 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 ...
13
votes
2answers
616 views

Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?

Let $X$ and $Y$ be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that $X$ embed in $Y$, and write $X \preceq Y$, if there exists a left invertible operator ...
13
votes
1answer
350 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 ...
13
votes
0answers
385 views

An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
13
votes
0answers
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 ...
12
votes
2answers
578 views

Point on a line nearest a point in Banach space

I have a Banach space geometry question (a curiosity-driven spin-off from a research topic). Given a point $x$ on the unit sphere of a Banach space and a vector $y\ne 0$, there is a multiple $t_0y$ of ...
12
votes
1answer
615 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 ...
12
votes
1answer
725 views

Banach spaces with few linear operators ?

Sometimes, dealing with the concrete and familiar Banach spaces of everyday life in maths, I happen nevertheless to ask myself about the generality of certain constructions. But, as I try to abstract ...
12
votes
0answers
1k 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 ...
11
votes
3answers
613 views

Continuous automorphism groups of normed vector spaces?

Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
11
votes
3answers
604 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
votes
2answers
869 views

“Orthogonal complement” of a subspace of a Banach space

I'm looking for a reference. I am considering the situation where $X$ is a Banach space and $Y$ is a closed finite co-dimensional subspace. I am looking for a $W$ that is a complement of $Y$ (i.e. so ...
11
votes
1answer
187 views

Containment of $c_0$

I have the following question. I guess it's quite simple for experts. Unfortunately, I could not come up with an answer yet. Let $X$ be a Banach space which contains no copy of $c_0$. Does it impply ...
11
votes
2answers
849 views

Approximating operators on Banach spaces by bounded operators on a proper dense subspace

While digging through old piles of notes and jottings, I came across a question I'd looked at several years ago. While I was able to get partial answers, it seemed even then that the answer should be ...
11
votes
1answer
271 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 ...
10
votes
6answers
2k views

Unbounded operator bounded in a dense subset

Let $X,Y$ be normed vector spaces, X infinite dimensional. Can I find 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 (i.e. $\sup _{x\in D, x \neq 0} ...
10
votes
3answers
2k 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 ...
10
votes
3answers
1k views

Projections in Banach spaces

Dear All, I am absolutely lost in the following problem: Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm ...
10
votes
1answer
408 views

Quotients of l^infty

Let $M$ be a closed subspace of $l^\infty$. Suppose that the quotient $l^{\infty}/M$ is isomorphic to $l^\infty$. Is it true that $M$ is complemented in $l^\infty$?
10
votes
1answer
613 views

Non-probabilistic proof of the Johnson–Lindenstrauss lemma

The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...
10
votes
1answer
349 views

Do ultrapowers of classical Banach spaces have unconditional bases?

I am trying to imagine (to some extent, of course) the geometry of ultrapowers of certain 'easy-to-handle' Banach spaces. Let me start with $X = \ell_p$, $p\in (1,\infty)$ or $X=c_0$. Since the ...
10
votes
2answers
298 views

Pull-back of generalized functions

Let $f\colon X\to Y$ be a smooth map between smooth manifolds. Then the pull-back operation $f^*\colon C^\infty(Y)\to C^\infty(X)$ is a linear continuous operator when $C^\infty$ is equipped with the ...
10
votes
0answers
268 views

Can we find arbitrarily many elements of SU(2) generating a good copy of MAX($\ell_1^n$) inside VN(SU(2))?

In trying to prove that the answer to the title is "no", I was led to the following problem (which I think is equivalent to the question asked in the title, but can be stated independently). If ...
9
votes
8answers
4k views

Can a self-adjoint operator have a continuous set of eigenvalues?

This should be a trivial question for mathematicians but not for typical physicists. I know that the spectrum of a linear operator on a Banach space splits into the so-called "point," "continuous" ...
9
votes
2answers
363 views

B(H) as a direct sum of a closed two sided ideal and a subalgebra

Let $B(H)$ is the C*-algebra of all bounded linear operators on Hilbert space $H$. Are there a closed two-sided ideal $I$ and a subalgebra $A$ of $B(H)$ such that $B(H)=I \oplus A$ (direct sum I and ...
9
votes
2answers
2k 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 ...
9
votes
3answers
439 views

Radii and centers in Banach spaces

Suppose I have a Banach space $V$ and a set $A \subseteq V$ such that for all $\epsilon > 0$ there exists $v$ such that $A \subseteq \overline{B}(v, r + \epsilon)$. Does there exist $c$ such that ...
9
votes
1answer
291 views

Inequivalent complete norms and the axiom of choice

Hi, I've been wondering about the following : Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space? All the examples of inequivalent complete norms ...
9
votes
1answer
851 views

Topological “Interpolation” ?

Let E be a normed space, and let $T$:E * $\rightarrow$ E * be a nonlinear operator. Suppose that : 1) $T$ is continuous from (E *, ||.||) to itself (i.e., it is norm-continuous). and 2) $T$ is ...
9
votes
1answer
308 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 ...