**11**

votes

**2**answers

927 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 ...

**6**

votes

**1**answer

599 views

### Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$

Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...

**9**

votes

**1**answer

392 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 ...

**62**

votes

**2**answers

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 ...

**25**

votes

**2**answers

2k 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 ...

**16**

votes

**5**answers

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

**30**

votes

**3**answers

2k views

### Absolute value inequality for complex numbers

I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
...

**24**

votes

**3**answers

2k 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 ...

**16**

votes

**4**answers

995 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.
...

**7**

votes

**2**answers

792 views

### Direct proof of injectivity of $L_\infty$

I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result.
$L_\infty$ as ...

**6**

votes

**1**answer

563 views

### “Vector bundle” with non-smoothly varying transition functions

I'm working my way through Lang's Fundamentals of Differential Geometry, and when he introduces vector bundles, he states that for finite dimensional bundles, the third axiom is redundant. I'm hoping ...

**14**

votes

**3**answers

609 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 ...

**12**

votes

**0**answers

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 ...

**10**

votes

**1**answer

461 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

**2**answers

441 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 ...

**4**

votes

**1**answer

120 views

### If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...

**8**

votes

**0**answers

256 views

### Closure of $L(\ell^2,\ell^2)$ in $L(\ell^2,\ell^\infty)$

Let $\ell^2$, $\ell^\infty$ denote the usual sequence spaces and let $L(\ell^2,\ell^2)$ the Banach space of bounded linear operators from $\ell^2$ to $\ell^2$ as well as $L(\ell^2,\ell^\infty)$ the ...

**4**

votes

**1**answer

217 views

### series representation in injective tensor products

All books on tensor products of Banach spaces contain the well-known theorem of Grothendieck that every element of the completed projective tensor product
$X \tilde{\otimes}_ \pi Y$ has a ...

**4**

votes

**0**answers

269 views

### Self-dual finite-dimensional complex normed spaces

Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space?
Remarks: There are easy counterexamples in the real case, and in ...

**5**

votes

**1**answer

282 views

### Info about Elton–Odell theorem

Hello everyone, could anyone please tell me where can I find information about the Elton–Odell theorem?
It states:
For any infinite dimensional Banach space $X$ there is a $q > 1$ so that $X$ ...

**4**

votes

**1**answer

299 views

### binary intersection property in finite dimension

I have recently discovered a survey article called "The Hahn-Banach Theorem: The Life and Times" by Narici and Beckenstein and I have read it with great amusement:
I was particularly fascinated by ...

**1**

vote

**0**answers

274 views

### Unambiguous “weak” vector valued $L^{+\infty}$ spaces?

For some time, I have been stuck to the problem to be described as follows. The (perhaps not so commonly known) facts given here are taken from R. E. Edwards' Functional Analysis (Holt, Rinehart and ...

**0**

votes

**0**answers

66 views

### Approximation Property: Characterization

Problem
Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$.
Suppose it has the approximation ...

**0**

votes

**0**answers

230 views

### Banach space of discontinuous functions on a product space

Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question.
For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a ...

**0**

votes

**0**answers

144 views

### On ultrafilters

Let $\mathcal{U}$ be a free ultrafilter of $\mathbb{N}$. Is true or false what $(C([0,1],\ell_p)^*)_{\mathcal{U}} = L_1(\mu, \ell_q)$, where $\mu$ is a measure, $C([0,1],\ell_p)^*$ is the dual of ...

**0**

votes

**1**answer

269 views

### Constructing a continuous matrix valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...

**0**

votes

**1**answer

253 views

### Integral in a σ−convex set.

Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...