**3**

votes

**0**answers

241 views

### Cameron Martin space

I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated.
1) It is the ...

**4**

votes

**1**answer

231 views

### Approximation of an integral over the unit ball of L_1

For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and
$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- ...

**1**

vote

**1**answer

65 views

### Does every $\alpha$-normal ordered Banach space have minimal upper bounds?

Let $\alpha>0$ and $X$ be an $\alpha$-normal (meaning, for $x,y\in X$,
$0\leq x\leq y$ implies $\|x\|\leq\alpha\|y\|$) ordered Banach space
with closed generating cone $X_{+}$. If $X$ is reflexive, ...

**4**

votes

**0**answers

116 views

### Hilbert metric of a sum of cones

Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...

**4**

votes

**1**answer

283 views

### extending compact operators to c0

Lindenstrauss has the following paper: http://www.ams.org/journals/bull/1962-68-05/S0002-9904-1962-10787-3/S0002-9904-1962-10787-3.pdf
I would like to see the proof for the following theorem (from ...

**8**

votes

**1**answer

263 views

### Uniqueness up to isometric isomorphism of predual of $(\sum_{\lambda\in\Lambda} H_\lambda)_{l_\infty}$ where $H_\lambda$ are Hilbert spaces

This fact is an easy consequence of results of the paper by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a simpler proof?

**4**

votes

**0**answers

465 views

### Do the banded operators check the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...

**5**

votes

**2**answers

255 views

### Contractively complemented subspaces of $c_0(I)$

Does every contractively complemented subspace of $c_0(I)$ is isometric to $c_0(J)$ for some $J\subseteq I$?
May be someone has a counterexample?

**6**

votes

**0**answers

240 views

### Is there Ultracoproduct-like construction for topological spaces in general?

In
http://arxiv.org/pdf/math/9704205.pdf
they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...

**5**

votes

**1**answer

131 views

### What Approximation Property does the space of Schatten-p class operators have?

Background
This is a follow-up question to:
What (classes of) Banach spaces are known to have Schauder basis?
In the previous question, I asked about what spaces are known to have Schauder basis. It ...

**4**

votes

**1**answer

319 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 ...

**4**

votes

**1**answer

219 views

### Weak convergence in the intersection of Lebesgue spaces or Sobolev spaces

Let $B:=B_1\cap B_2\cap...\cap B_n$, where each $B_j$ is a reflexive Lebesgue space or Sobolev space (such as $L^4$, $H^1$, etc.) on a domain in $\mathbb{R}^d$. Then $B$ is a Banach space endowed with ...

**1**

vote

**1**answer

203 views

### Agreement of two topologies on a linear space

I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.
Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...

**11**

votes

**1**answer

439 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 ...

**7**

votes

**2**answers

1k views

### Direct proof of the separation theorem of Hahn-Banach

The "extension" (or "analytic") form of the theorem of Hahn-Banach has a natural and yet elegant proof. In just any textbook I have ever seen, it is proved first; the "separation" (or "geometric") ...

**4**

votes

**1**answer

221 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 ...

**3**

votes

**1**answer

196 views

### Continuity of lattice operations in Banach lattices

Let $L$ be a Dedekind-complete Banach lattice. Let $\mathcal{B}$ be the family of nonempty norm-compact subsets of $L$ that are bounded from below. Endow $\mathcal{B}$ with the topology induced by ...

**3**

votes

**0**answers

68 views

### $x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology

This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...

**14**

votes

**1**answer

657 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 ...

**5**

votes

**0**answers

155 views

### almost projective Banach space, complex scalars

It is well-known that if a real Banach space $E$ is "almost metrically projective" then $E$ is isometrically isomorphic to some $\ell^1(\Gamma)$. We say $E$ is "almost metrically projective" if ...

**1**

vote

**0**answers

83 views

### Is scalarwise measurability determined by the strong dual?

Since this question has not received an answer so far, I try to reformulate the question in a simpler manner as follows: Do there exist $E,F,\ell,f$ such that
$E$ and $F$ are separable (real) ...

**8**

votes

**3**answers

753 views

### Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$

Hi.
Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I mean by this ...

**4**

votes

**0**answers

94 views

### Categorical notions involving $\ell_p$ spaces.

First of all, apologies for a somewhat vague question but let me give a try. We know what the projective objects in the category of Banach spaces are: these are precisely $\ell_1(\Gamma)$-spaces. (One ...

**1**

vote

**1**answer

131 views

### Special kind of operators

Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation
$$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$
where ...

**4**

votes

**3**answers

391 views

### radon-nikodým property of $\ell^\infty$

I am wondering whether $\ell^\infty(\mathbb N)$ has the Radon-Nikodým property. Of course $\ell^1(\mathbb N)$ does, but I was unable to find out whether (e.g.) duals of spaces with the R-N property ...

**2**

votes

**2**answers

238 views

### Martingale-cotype vs cotype on super-reflexive spaces

I'm have difficultly nailing down the direction of some implications. For $2 \leq q < \infty$, there are (at least) two ways to say that a Banach space $B$ has "cotype $q$".
$B$ has cotype q.
...

**3**

votes

**1**answer

107 views

### What is the doubling dimension of convex functions?

I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the ...

**2**

votes

**2**answers

171 views

### Are compact sets in a Banach lattice order bounded?

Given a compact subset $A$ of a Banach lattice $E$, is the following true?
There exist $u,v\in E$ so that $u\leq a\leq v$ for all $a\in A$.
This is true in case $E=C(X)$, $X$ compact, with the ...

**0**

votes

**1**answer

188 views

### Extensions of Carathéodory's theorem

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex ...

**0**

votes

**0**answers

134 views

### isomophism, commutator

Let X be a Banach space. $B(X)$ is the algebra of all bounded linear operators on X.
$\phi: B(X)\rightarrow B(X)$ is a isomoprphisn, $\varphi: B(X)\rightarrow B(X)$ is a isomorphism or negative ...

**2**

votes

**0**answers

246 views

### Finite codimensional subspaces of L(X,Y)

Let $X$ and $Y$ be separable Banach spaces and $L(X,Y) $ be the Banach space of bounded linear operators from $X$ to $Y$. Suppose $A$ is a norm closed finite codimensional subspace of $L(X,Y)$.
My ...

**3**

votes

**2**answers

305 views

### On hyperplanes of $L\infty$

Consider the hyperplane $H=\{f\in L^\infty: \int f = 0\}$ of $L^\infty = L^\infty[0,1]$. My question is:
1. What is the Banach-Mazur distance between $H$ and $L^\infty$? Are there "natural" ...

**0**

votes

**0**answers

138 views

### Gaussian width (or metric entropy) for the intersection of the $\ell_1$ and $\ell_2$ balls

Let $B_p := \{ x \in \mathbb{R}^d:\; \|x\|_p \le 1\}$ where
$\|x\|_p := (\sum_{i=1}^d |x_i|^p)^{1/p}$ is the $\ell_p$ norm.
(1) Let $t \in (0,1)$. Can we give an estimate on $$\mathbb{E} ...

**4**

votes

**3**answers

613 views

### Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions?

Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...

**10**

votes

**2**answers

485 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

**0**answers

270 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 ...

**4**

votes

**3**answers

182 views

### Quotients with unconditional bases

Gowers' dichotomy theorem asserts that every Banach space either contains an HI subspace or a subspace having an unconditional basis. There are examples of HI spaces without quotients having ...

**1**

vote

**1**answer

421 views

### Is this result of Spain correct?

Let us have a look on the proof of Theorem 2 in [P. G. Spain, Boolean algebras of projections, Proceedings of the Edinburgh Mathematical Society (Series 2) 19, 03, March 1975, 287-289]
The author ...

**9**

votes

**2**answers

622 views

### Complexifying a real Banach space and its dual

A standard way to define the "complexification" $E_\mathbb{C}$ of a real Banach space $E$ is to define a complex linear structure on $E\times E$ by (1) $(x,y)+(u,v)=(x+u, y+v)$, (2) ...

**1**

vote

**0**answers

49 views

### Extension of $S_+$ type operators

Let $X$ be a reflexive Banach space and $G\subset X$ a open bounded set. Let $F:\overline{G}\rightarrow X^\star$ be a $S_+$ operator, i.e., if for any sequence $x_n$ in $G$ for which ...

**5**

votes

**0**answers

166 views

### Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...

**2**

votes

**2**answers

330 views

### Banach lattice subspace of $C([0,1])$ not a sublattice

This is probably easy, but I did not see it in standard texts. Describe a closed subspace $V$ of $C([0,1])$ such that $V$ is a Banach lattice (in the pointwise ordering), but $V$ is not a sublattice ...

**9**

votes

**1**answer

317 views

### Subspaces of $l_p$ and Banach-Mazur distance

This is a question I posted on SE, and I have been advised to post it here.
http://math.stackexchange.com/questions/146427/subspaces-of-l-p-and-banach-mazur-distance
It is well-known that every ...

**2**

votes

**1**answer

270 views

### Decomposing bilinear forms in Hilbert spaces

You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...

**0**

votes

**1**answer

352 views

### Norms agreeing on dense subspace [closed]

Suppose $(B,\|\cdot\|)$ is a Banach space, $V\subset B$ a dense subspace, and $V$ is equipped with a norm $\|\cdot\|_V$ such that $\|x\|_V = \|x\|$ for all $x\in V$.
Is $(B,\|\cdot\|)$ a completion ...

**3**

votes

**0**answers

429 views

### question about bidual normed space

Consider a Banach $\mathbb{R}$-space E and an element $u\in E''$ such that :
for all sequence $\phi_n\in E'$ which $\sigma(E',E)$-converges to $\phi \in E'$, one has $\lim u(\phi_n)=u(\phi)$.
Is it ...

**0**

votes

**1**answer

211 views

### Absolute norms and 1-unconditional sums

Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, ...

**5**

votes

**1**answer

309 views

### Non-super reflexive space

Suppose $X$ is a reflexive space (possibly non-separable) which is not super-reflexive. Then (by definition) there exists a non-reflexive Banach space $Y$ which is non-reflexive but is finitely ...

**7**

votes

**0**answers

267 views

### Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...

**4**

votes

**1**answer

151 views

### Generator of a $C_0$-semigroup restricted to a subspace

Suppose we have a decreasing filtration of Banach spaces $(E_t)\_{t\geq0}$, inclusions $V_{s+t,t}:E_{s+t}\to E_t$ and projections $P_{t,s+t}:E_t\to E_{s+t}$ such that $P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}$. ...