The banach-spaces tag has no wiki summary.

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### 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**

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**0**answers

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### 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

**1**answer

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**

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**0**answers

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### 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**

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**2**answers

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### 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**

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**3**answers

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### 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**

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**2**answers

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### 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**

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**7**answers

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### 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

**2**answers

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**

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**5**answers

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### 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 ...

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**5**answers

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### 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 $\frac{1}{N}$:
$$ ...

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**4**answers

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.
...

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**0**answers

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**

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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**

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**5**answers

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 ...

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**3**answers

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### 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**

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**1**answer

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 ...

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**1**answer

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 ...

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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 ...

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**3**answers

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### 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 ...

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**2**answers

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 ...

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### 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 ...

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**1**answer

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### 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 ...

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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? ...

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### 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 ...

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**2**answers

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 ...

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**1**answer

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 ...

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**1**answer

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### 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 ...

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### 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 ...

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**3**answers

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 ...

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### 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 ...

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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 ...

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**1**answer

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 ...

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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 ...

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**1**answer

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 ...

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**6**answers

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### 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} ...

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**3**answers

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### 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 ...

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### 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 ...

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**1**answer

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 $l^\infty$?

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613 views

### Non-probabilistic proof of the Johnsonâ€“Lindenstrauss lemma

The Johnsonâ€“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 ...

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**1**answer

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 ...

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**2**answers

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 ...

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**0**answers

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 ...

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### 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" ...

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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 ...

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### 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 ...

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**3**answers

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 ...

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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 ...

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**1**answer

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 ...

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**1**answer

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 ...