A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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111 views

### Interpolation of some Lebesgue spaces

When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...

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**1**answer

234 views

### Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?

The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...

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56 views

### Separable measures on compact groups

Let us say that a (signed, finite) measure $\mu$ is separable if $L_1(|\mu|)$ is a separable Banach space.
EDIT: Suppose that $G$ is a locally compact group such that each measure on $G$ is ...

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**1**answer

147 views

### The continuum hypothesis and the diamond principle for $\aleph_1$

In [S. Shelah. Uncountable constructions for B.A. e.c. groups and Banach spaces. Israel J. Math. 51 (1985), 273-297], the existence of a special Banach space is proved, assuming the diamond principle ...

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82 views

### Dual of the space of affine functions

Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\...

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139 views

### Compact images of nowhere dense closed convex sets in a Hilbert space

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$.
Question. Is there a non-compact linear bounded operator ...

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28 views

### A fixed point in orderd cone

While studing the fixed point in orderd fixed point I noticed that most of the time they worked in a Banach space orderd by a cone's order:
Let $X$ be an ordered Banach space with order $\leq $. A ...

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269 views

### Is there a version of Fischer-Riesz theorem for Banach space?

$( \Omega,F, P )$: a measurable space equipped with a finite measure
$(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra
$p$ : a constant bigger than $1$
...

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**1**answer

111 views

### Is it true that every Banach space has at least one extreme point that is normed by some point?

Definition: Let $X$ be a Banach space and $X^*$ be its continuous dual of $X,$ that is, $X^*$ contains all bounded linear functionals on $X.$
Denote
$$B_{X^*} = \{x^*\in X^*: \|x^*\|_{X^*}\leq 1\}.$$...

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107 views

### Complemented subspaces of $C(\beta\mathbb N\times \beta\mathbb N)$

Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\...

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115 views

### A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters

Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$?
(...

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**1**answer

145 views

### Counter example about blow-up solution of DEs

Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...

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167 views

### About finite representability of Banach space

Can someone please tell me the brief sketch (or any known reference) of the following results?
Why $\ell_2$ is finitely representable in any infinite-dimensional Banach space?
Why every Banach space ...

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**1**answer

92 views

### Is the norm of the Banach space projective tensor product of finite-dimensional C*-algebras a C*-norm?

Let $A$ and $B$ be two finite-dimensional C*-algebras.
Let $\gamma$ denote the projective Banach space tensor product norm on the algebraic tensor product $A\odot B$, so $\gamma(t)=\inf\{\sum_{i}\|...

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438 views

### Multiple of identity plus compact

Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of ...

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**1**answer

221 views

### Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?

Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$
is not ...

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**1**answer

83 views

### Optimal estimate in trace norm

Let $x,y$ be vectors of some Hilbert space of unit length.
Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$
Assume then that we know that $\left\lVert x-...

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**0**answers

88 views

### Fixed-point properties for affine actions of topological groups

T. Mitchell [Illinois J. Math. 14 (1970) 630--641] defined four properties of a topological semigroup, and in particular of a topological group $G$.
Two of them are:
(F2) Every jointly continuous ...

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**1**answer

253 views

### Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional.
Question. Does it follow that $A$ is finite dimensional?
...

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291 views

### Does point-wise weak convergence give weak convergence in $L^2(I;X)$?

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...

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82 views

### Orthogonal complement vector space

Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study
$X^{\perp_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle_{L^2} =0 \ \forall x \in X \right\}$
and
$X^{\perp_{H^{-...

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177 views

### Which subspaces of $\ell_p^n$ are isometric?

This question is similar to the one asked here:
Extending linear isometries from subspaces of $\ell_p^n$
Let $p$ be an even integer. Let $X,Y$ be subspaces of $\ell_p^n$, and let $U : X \to Y$ be a ...

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419 views

### On a special type of normed linear spaces

Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying
$$
\|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z}
$$ is a group ...

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votes

**1**answer

102 views

### Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2
Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...

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**0**answers

58 views

### Nonlinear maps in Riesz Thorin theorem

The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...

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votes

**1**answer

84 views

### Why ker(1) is a semi M-ideal in $\ell_1$?

A subspace $Y$ of $X$ is said to be semi M-ideal if $\exists$ a projection $P$ (not necessarily linear) from $X^*$ to $Y^\perp$ such that $\|x^*\|=\|Px^*\|+\|x^*-Px^*\|$. And also, $P(\lambda x^*+Py^*)...

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551 views

### If $E\oplus_\phi E \cong E\oplus_\psi E,$ does it imply that $\phi= \psi$?

Let $E\neq \{0\}$ be a Banach space.
For each $p\in[1,\infty), $ we define
$$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$
Let $F$ be another Banach space.
By $E\...

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votes

**1**answer

98 views

### Does there exists an extreme point $(a_1^*,…,a_n^*)$ of $B_{\mu^*}$ such that $a_i^*\neq 0$ for all $1\leq i\leq n$ and $\sum_{I=1}^n a_i^*a_i=1?$

Fix a natural number $n\geq 1.$
Let $\mu$ be a norm on $\mathbb{R}^n$ satisfying
$$\mu(0,...,0,\stackrel{i}{1},0,...,0) = 1 \quad\text{for all }1\leq i\leq n.$$
Let
$$B_{\mu} = \{(a_1,...,a_n)\in \...

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**1**answer

105 views

### Do functions exist and are they dense? Or does it depend on the basis?

Consider an orthonormal basis $(\varphi_n)_{n \in \mathbb N}$ of $L^2(\mathbb R).$
We consider the functionals $\Phi_n$ given by $$ C^b(\mathbb R) \ni f \mapsto \left\langle \varphi_n, f \varphi_{n+1}...

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**0**answers

41 views

### The dual relationships between strictly singular operators and strictly cosingular operators

Recall that an operator $T:X\rightarrow Y$ between Banach spaces is called strictly singular if if it is not an isomorphism when restricted to any
infinite-dimensional closed subspace of $X$. An ...

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votes

**1**answer

139 views

### Approximation property counterexamples? (Also: relation to tensor products)

I remember reading somewhere (but unfortunately, I've forgotten where it was) that the canonical map from the (completed) projective tensor product of two Banach spaces to the (completed) injective ...

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**1**answer

94 views

### A formula for vector valued measurable functions

Let $B_{\infty}(\Omega)$ be the space of bounded measurable functions on the measurable space $\Omega$. For a given Banach space $X$, let us denote $B_{\infty}(\Omega,X)$ by the set of all bounded ...

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**1**answer

105 views

### Compactness of operators and norming sets

Originally asked on MSE.
Let $T$ be a linear map from a normed space $E$ into a Banach space $F$.
Let $D\subset \overline{B}_{F^{\ast}}$ be norming, i.e., there is $r>0$ such that $\sup\limits_{v\...

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vote

**1**answer

45 views

### Infinitely many independent functions that are only frequency localized?

A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...

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votes

**1**answer

144 views

### Central limit theorem in Banach space in scheme of series

I wonder whether Theorem 2 from the paper J. Zinn, Annals of Probability, 1977, vol. 5, 283-286 can be extended to the CLT for a scheme of series. (The paper is available in the web.)
Let $G$ be ...

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**1**answer

148 views

### Does Bishop-Phelps Theorem hold for extreme points (slightly different version)?

Recall the Bishop-Phelps Theorem.
Bishop-Phelps Theorem: Let $B\subseteq E$ be a bounded, closed, convex subset of a real Banach space $E.$
Then the set
$$\{e^*\in E^*: e^* \text{ attains its ...

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**3**answers

142 views

### Preannihilators of subspaces of separable duals

If $Y\subset X^*$ is a closed subspace (where $X$ is a separable Banach space), the preannihilator of $Y$ in $X$ is $Y_{\perp}:=\{x\in X : y^*(x)=0, \forall y^*\in Y \}$. If $Y$ is a proper subspace ...

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**1**answer

133 views

### Does separability of the strong operator topology imply separability of the underlying space?

Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.
Suppose that the strong operator topology on $B(X)$ is separable and that the cardinal number of $B(X)$ is continuum.
...

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**2**answers

279 views

### If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?

Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?

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**1**answer

75 views

### Norming functionals for vectors in intersections

Suppose that $(X, \|\cdot\|_X)$, $(Y, \|\cdot\|_Y)$ are two Banach spaces such that $X\subset Y$ and $\|x\|_Y\leq \|x\|_X$ for all $x\in X$ and $X$ is dense in $(Y, \|\cdot\|_Y)$.
Every functional $...

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**1**answer

153 views

### Relation between the weak star topology and hereditary Lindelöfness

Let $X$ be a Banach space. Is the following implication valid?
$$ (X,w) \textrm{ is hereditarily Lindelöf}~ \Rightarrow X^*~ \textrm{is separable} $$
The converse is clearly true, since the ...

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**1**answer

129 views

### Lipschitz right inverses of Banach space quotients

Let $X$ be a Banach space and $Y$ a closed subspace of $X$. I am interested in quotients $q:X\to X/Y$ that do not have Lipschitz right inverses (not necessarily linear).
Of course, if $Y$ is ...

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**2**answers

185 views

### pointwise convergence to the identity

Let $X$ be a separable topological vector space with size (cardinal number) no larger than $\mathfrak{c}$. Does there exist any sequence of finite rank linear maps $\phi_n:X\to X$ pointwise converging ...

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**0**answers

81 views

### point-wise approximation of the identity in hereditary Lindelof spaces

Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery ...

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**1**answer

191 views

### Subspaces isomorphic with quotients

Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?

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136 views

### Is this property an isomorphic characterization of $\ell_1(\Gamma)$?

Let $\Gamma$ be an infinite set. Then every $(x_i)_{i\in\Gamma}\in \ell_1(\Gamma)$ has at most a countable number of components $x_i\neq 0$.
As a consequence, every separable subspace $M$ of $\ell_1(\...

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**1**answer

295 views

### Are all compact subsets of Banach spaces small in a measure-theoretic sense?

Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...

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**0**answers

135 views

### Status of an open problem in isometric aspect of Banach space theory

The following open problem is taken from the book Open Problems in the Geometry and Analysis of Banach Spaces, page $40.$
Problem $84:$ Assume that $X$ is an infinite-dimensional separable Banach ...

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votes

**3**answers

469 views

### $c_0$ is not isometrically isomorphic to $c$

Let us consider the space of convergent sequences which is denoted by $c$. The space of all sequences $(x_n)\in c$ with $\lim x_n=0$ is also denoted by $c_0$. Clearly $c_0$ is a proper closed ...

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**1**answer

132 views

### A particular separation example

Q1. Does there exist a separable Banach space $X$ satisfying in the following property?
1- $X^*$ is non separable.
2- For every countable subset $F\subset X^*$ there exists $0\neq x_F\in X$ ...