**3**

votes

**1**answer

106 views

### almost invariant half space for a dual of a restricted operator

Let $X$ be an infinite-dimensional Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. A closed subspace $Y$ of $X$ is said to be an almost-invariant halfspace (...

**1**

vote

**0**answers

44 views

### Linear independence of an odd set of measurable functions

Let $g(t)$ be a convex positive function. I'm trying to show that the set $\{ |t|^ne^{\frac{g(t)}{t^n}}\}_{n\in \mathbb{N}}$ is linearly independent in the space of measurable functions on $\mathbb{R}...

**2**

votes

**2**answers

343 views

### Elements of Minimal Norm on an affine subset of a Banach Space

Let $v_1$ and $v_2$ be two elements of a real Banach space $(X,\lVert\;\rVert)$, each of unit norm. Consider the one-dimensional affine subspace $v_1+tv_2$ for real $t$. Is there a general formula for ...

**1**

vote

**1**answer

49 views

### Bessaga-Pelczynski Selection Principle

Bessaga-Pelczynski Selection Principle states that if $(x_{n})_{n}$ is a basis for a Banach space $X$, then every normalized weakly null sequence $(y_{n})_{n}$ in $X$ admits a subsequence that is ...

**7**

votes

**0**answers

97 views

### Choice and the Baire property in non-separable complete metric spaces

It's known to be consistent with ZF+DC that every subset of $\mathbb{R}$ has the Baire property (BP). (E.g. Shelah's model). If so, then every subset of every complete separable metric space has ...

**15**

votes

**1**answer

563 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{...

**7**

votes

**1**answer

375 views

### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
$...

**5**

votes

**0**answers

92 views

### Banach spaces complemented in their ultrapowers

By the principle of local reflexivity, the second dual $X^{**}$ of a Banach space $X$ is complemented in some ultrapower $X^U$ of $X$. Even when $X$ is separable, the index set of $U$ cannot be ...

**1**

vote

**1**answer

67 views

### Complemented subspaces in the James space

Let $J$ be the James space. I have the following questions:
Question 1: Does every infinite-dimensional closed subspace of $J$ contain an infinite-dimensional closed subspace that is $C$-complemented ...

**3**

votes

**1**answer

217 views

### A unital algebra with norm and continuous multiplication is a Banach algebra

In my research in functional analysis, I came across this rather simple result:
For a normed algebra A over $ \mathbb{C} $ with unit, in which multiplication , right and left are both continuous w....

**10**

votes

**2**answers

366 views

### Extracting subsequences in Banach spaces, along an ultrafilter?

There are various principles in Banach space theory that allow one to pass from a given sequence of vectors $(x_n)$, to a subsequence $(x_{n_k})$ with some desired property. I'm thinking here, in ...

**5**

votes

**0**answers

120 views

### Norm of projection onto functions of mean zero

Let $X$ be a finite set and consider the space $\ell^2(X;Y)$ of functions $\zeta:X\to Y$, where $Y$ is a fixed Banach space. It decomposes into a direct sum of constant function and its complement $\...

**2**

votes

**1**answer

352 views

### Is the $L^1$-space dual to a Banach space

Let $(\Omega,\mu)$ be a measure space. It is well known that for $1<p\leq \infty$ one has the duality
$$L^p=(L^{p*})^*,$$
where $1/p+1/p^*=1$.
Question. Is it known that the Banach space $L^1$ is ...

**0**

votes

**1**answer

234 views

### Normed space between $H^{0+}$ and $L^2$

In the space $\in L^2(\mathbb{R}^3)$, consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$
Of course if $f\in H^s(\mathbb{R}^3)$ ...

**0**

votes

**0**answers

69 views

### Weak Topology and Domain Theory: Which topology on the function domain restricts to the weak topology on C([0,1])?

Let $\mathbb{IR}$ be the interval domain over the set $\mathbb{R}$ of real numbers, defined by:
$$\mathbb{IR} := \{ [a,b] \mid a, b \in \mathbb{R}, a \leq b\} \cup \{ \mathbb{R}\},$$
and ordered by ...

**2**

votes

**1**answer

78 views

### Sequences in $L_{p}(1<p<\infty)$ that is equivalent to the unit vector basis of $l_{p}$ or $l_{2}$

Let $1<p<\infty$. Johnson and Schechtman (Multiplication operators on $L(L_{p})$ and $l_{p}$-strictly singular operators, 2008, DOI: 10.4171/JEMS/141, eudml, arxiv) observed that if $(x_{n})_{n}$...

**2**

votes

**0**answers

99 views

### A universal operator between separable Banach spaces

The Banach space $C[0,1]$ is universal for all separable Banach spaces in the sense that for a separable Banach space $X$ there is an isometric isomorphism from $X$ into $C[0,1]$. My question is ...

**3**

votes

**1**answer

115 views

### Subspaces of $L_{p}(2<p<\infty)$

Let $p>2$ and $X$ a subspace of $L_{p}$.
Then Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$.
Question: if $X$ is ...

**0**

votes

**1**answer

50 views

### Weakly null sequences in $L_{p}(1<p<2)$

Let $1<p<2$. Let $(f_{n})_{n}$ be a normalized weakly null sequence in $L_{p}$ such that the sequence $(f_{n})_{n}$ contains no subsequence that is equivalent to the unit vector basis of $l_{p}$....

**3**

votes

**1**answer

139 views

### Reference: Finsler Derivative?

On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...

**2**

votes

**0**answers

150 views

### Banach spaces: A ball being a subset of the interior of the union of two balls

Let $X$ be a separable reflexive Banach space and let $A$, $B_1$, and $B_2$ be three closed balls in $X$. Is there a `handy' necessary and sufficient condition for checking whether $A \subseteq (B_1 \...

**4**

votes

**0**answers

76 views

### Equivalence of questions regarding restrictions of pure states

In Davidson and Szarek's article "Local Operator Theory, Random Matrices and Banach Spaces" in the Handbook of the Geometry of Banach Spaces, the authors discuss the (now solved) Kadison-Singer ...

**1**

vote

**1**answer

200 views

### When is the sum of complemented subspaces complemented?

Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...

**0**

votes

**1**answer

81 views

### When do two quasi-Banach spaces with identical dual spaces have equivalent norms?

Let $X$ and $Y$ be two quasi-Banach spaces such that the dual spaces satisfy $X^*=Y^*$.
I want to know if there are some conditions that imply $X=Y$ (in the sense of equivalent norms).

**5**

votes

**1**answer

346 views

### Renorming a Banach space to make projections contractive

Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$.
Can the same be done for a family of projections? That is, given finitely many ...

**2**

votes

**0**answers

69 views

### Self-adjoint, strictly singular operators on Hilbert spaces

Let $X$ and $Y$ be infinite-dimensional Banach spaces. Recall that an operator $T: X\rightarrow Y$ is strictly singular if it is not an isomorphic embedding when restricted to any infinite-dimensional ...

**3**

votes

**0**answers

81 views

### Products of strictly singular operators on $L_{p}[0,1]$ or on $C[0,1]$

In 1970, V.D. Milman (Operators of class $C_{0}$ and $C^{*}_{0}$, Teor. Funkc. Funkc. Anal. Ih Priloz. 10(1970),15-26) proved that the product of two strictly singular operators on $L_{p}[0,1](1\leq p&...

**18**

votes

**2**answers

735 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 ...

**8**

votes

**1**answer

256 views

### Interpolation between $L_1^0$ and $L_2^0$

Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...

**3**

votes

**2**answers

183 views

### On the Lorentz sequence space $d(w,1)$

I am interested in examples of dual Banach spaces $X$ with the Schur property (weakly convergent sequences in $X$ are norm convergent) like $\ell_1$.
The Lorentz spaces $d(w,1)$ [Lindenstrauss and ...

**4**

votes

**1**answer

419 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 ...

**5**

votes

**0**answers

69 views

### Set of w*-continuous operators closed for the weak* topology or not?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...

**3**

votes

**0**answers

98 views

### quasi-nilpotent part of a dual operator

Definitions and notation.
Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as
\begin{equation*}H_0(T):=\left\{...

**7**

votes

**1**answer

92 views

### About $C(K)$-spaces containing no copy of $l_{1}$

Let $K$ be a compact Hausdorff space. I wonder whether there are characterizations of $K$ such that $C(K)$ contains no copy of $l_{1}$. There are some compact Hausdorff spaces $K$ such that $C(K)$ ...

**0**

votes

**0**answers

110 views

### Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$

Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...

**8**

votes

**1**answer

196 views

### Davis, Figiel, Johnson and Pełczyński factorization through spaces with a bases

Davis, Figiel, Johnson and Pełczyński's Factorization Theorem states that each weakly compact operator $T:X \to Y$ between Banach spaces $X$ and $Y$ factors through a reflexive Banach space $Z$. In ...

**0**

votes

**1**answer

88 views

### Sequentially continuous but not continuous linear map $(X^*, w^*)$ to $(Y^*,w^*)$

Let $X, Y$ be Banach spaces and let $T : (X^*, w^*) \rightarrow (Y^*,w^*)$ be a linear map. Suppose that $T$ is sequentially continuous. Must $T$ be weak*-to-weak*-continuous ?

**0**

votes

**0**answers

80 views

### Subprojectivity of the spaces $B_{p}(1<p<\infty)$

Fix $1<p<\infty$. In his thesis, C.J.Seifert adapted the construction of Baerstein's space $B$ by replacing the number 2 by $p$ to construct the space, called "$B_{p}$" space. He showed that ...

**1**

vote

**1**answer

64 views

### Subprojectivity of $L_{p}(p>2)$

Let $p>2$. Following from M.I. Kadec and A. Pełczyński's results (Studia Math. 1962), R.J.Whitley (Trans. Amer. Math.Soc. 1964) observed that $L_{p}$ is subprojective, that is, every infinite-...

**3**

votes

**1**answer

107 views

### On the complemented subspaces of $L_{p}(p>2)$

M.I. Kadec and A. Pełczyński proved that if $E$ is a subspace of $L_{p}(p>2)$ isomorphic to $l_{2}$, then $E$ is complemented in $L_{p}$. My question is:
Is there a constant $C_{p}$ depending only ...

**2**

votes

**0**answers

74 views

### Reflexive subspaces of dual spaces

If $X$ is a Banach space, is it true that $X^{*}$ must contain an infinite dimensional reflexive subspace? I know of Gowers' example of a Banach space not containing $c_0$, $l_1$, or reflexive, but I ...

**5**

votes

**2**answers

147 views

### Biorthogonal functionals

If $X$ is a separable Banach space and $(x_n)$ is a basic sequence, then we can define biorthogonal functionals $(x^{*}_n)$ in $X^{*}$ such that $x^{*}_n(x_k)=\delta_{nk}$.
What about conversely? If ...

**4**

votes

**1**answer

113 views

### Trivial intersection of kernels

This is a follow up to the question: Biorthogonal functionals. A positive answer to that question implies a negative answer to this one.
If $X$ is a separable Banach space, can we find a basic ...

**3**

votes

**1**answer

55 views

### Dual of colimit in $\text{Ban}_1$

I learned in J. Castillo's Hitchhiker guide to categorical Banach space theory that, by a theorem of Semadeni and Zidenberg, limits and colimits exist in the category $\text{Ban}_1$ of Banach spaces ...

**2**

votes

**1**answer

104 views

### A single point as sum range of a series

Can anyone give some clues to show that any infinite dimensional Banach space have a conditionally convergent series whose sum range is a single point?
Thanks in advance for any help.

**2**

votes

**1**answer

187 views

### Extension of a function from almost everywhere to everywhere

The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points?
Example: Let $f(x)=\...

**8**

votes

**1**answer

220 views

### Non-reflexive Orlicz spaces

I am looking for a good reference on Orlicz spaces; I would appreciate any books treating this topic from the Banach-space perspective. For example, I would like to find a reference to the following ...

**2**

votes

**1**answer

114 views

### Two questions on the James $p$-space $J_{p}(1<p<\infty)$

Let $1<p<\infty$. The James $p$-space $J_{p}$ is the Banach space of all sequences of real numbers $(a_{i})_{i}\in c_{0}$ such that
$$\|(a_{i})_{i}\|=\sup\{(\sum_{j=1}^{n}|a_{p_{j-1}}-a_{p_{j}}|...

**3**

votes

**1**answer

101 views

### A question on the quantification of compact operators

Let me first fix some notations.
If $A$ and $B$ are nonempty subsets of a Banach space $X$, we set $$d(A,B)=\inf\{\|a-b\|:a\in A,b\in B\},$$$$\widehat{d}(A,B)=\sup\{d(a,B):a\in A\}.$$
Let $A$ be a ...

**23**

votes

**1**answer

2k 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?