**1**

vote

**0**answers

75 views

### Dual of pre-dual of BV

Was there any relevant work to determine the dual (or more likely the predual) of the space of bounded variation functions $BV(\mathbb{R}^n)$ (I recall the definition : a function in ...

**-4**

votes

**0**answers

111 views

### show that L(X,Y)banach then Y banach

Let {Xα}α∈A be a collection of Banach spaces. It is easy to show that P={(xα):supα∥xα∥<∞} with ∥(xα)∥=supα∥xα∥ is a banach space.
If the indexing set A is finite, then it is easy to show that P ...

**0**

votes

**2**answers

83 views

### Book and Papers for properties of uniformly convex and locally uniformly convex and strictly convex Banach spaces.

I am looking for reference books and research articles which cover analysis of uniformly convex and locally uniformly convex and strictly convex Banach spaces.

**7**

votes

**1**answer

306 views

### Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?

The Open Mapping Theorem, the Bounded Inverse Theorem, and the Closed Graph Theorem are equivalent theorems in that any can be easily obtained from any other. The Closed Graph Theorem also easily ...

**2**

votes

**1**answer

247 views

### Banach space of discontinuous functions(Killing continuous functions)

Edit: According to the comment of Prof. Majer, I revise the question:
For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$
...

**3**

votes

**1**answer

241 views

### When $C(X)$ is an injective $C(X)$-module?

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space.
I would like to know if the weakened module version of this question is answered. More precisely: ...

**4**

votes

**1**answer

123 views

### Can one show that the dual of a quasi-Banach space separates points without explicitly identifying the dual?

I'm interested in a question regarding the identification of some duals of quasi-Banach spaces.
However, I'm not familiar with the quasi-Banach literature, so I'm hoping somebody can point me in the ...

**10**

votes

**2**answers

463 views

### Banach space modulo a one-dimensional subspace =?

My question is the following:
Given an infinite dimensional Banach space $E$ and a one-dimensional linear subspace $F\subset E$. It is well-known that this one-dimensional linear subspace is closed ...

**5**

votes

**1**answer

130 views

### Reflexive subspaces of non-separable abstract $L_1$ spaces

An abstract $L_1$ space is a Banach lattice $E$ such that $\|x+y\|=\|x\|+\|y\|$ for disjoint $x,y\in E$. The space $L_1[0,1]$ is a separable example that contains subspaces isomorphic to $L_p[0,1]$ ...

**17**

votes

**0**answers

245 views

### Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set ...

**3**

votes

**4**answers

296 views

### Continuity in Banach space for non-linear maps

I want to find an example of a Banach space $X$ and a continuous map $f:X\rightarrow X$ such that $f$ is not bounded on the unit ball. I do not doubt that such an example exists, but I cannot make it ...

**2**

votes

**1**answer

111 views

### Necessary conditions for optimality in Banach spaces

Let $X$ denote the non-negative "orthant" of the Banach space $L^2$ (or whatever you call the set of functions in $L^2$ that are non-negative), and let $C$ be a closed, convex subset of $X$. Let $f$ ...

**9**

votes

**1**answer

601 views

### Counterintuitive consequences of the Hahn-Banach theorem

The axiom of choice has many counterintuitive consequences like the Banach-Tarski paradox.
The Hahn-Banach theorem is a consequence of the axiom of choice, but it is weaker.
I would like to know ...

**0**

votes

**1**answer

161 views

### Aronszajn measure

In a geometric measure theory (GMT) course I'm following this year, the professor told us about the Aronszajn measure, and asked us to go check by ourselves what it reprensents (the course was about ...

**1**

vote

**1**answer

161 views

### James $\ell_1$-theorem

This question was asked at MSe before but with no answer.
I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's ...

**3**

votes

**3**answers

485 views

### What should be considered a finite size of an infinite dimensional space? [closed]

I've got a map between two infinite dimensional spaces, $f: A\to B$, where $A$ seems "larger" than $B$. For the sake of conversation let's assume that $A$ is the set of smooth maps $\mathbb R^3\to ...

**2**

votes

**2**answers

264 views

### If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Pietro Majer ...

**3**

votes

**2**answers

148 views

### A lower bound on $\|\cdot\|_{p_{\ast}}$ image of $\ell^{q_{\ast}}$ vectors

The motivation for this question comes from this paper where we study lower bounds on the complexity of convex optimization algorithms (BTW, if you find a better title for my question, please let me ...

**4**

votes

**1**answer

110 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 ...

**3**

votes

**1**answer

83 views

### Extreme unit linear functional not norming a vector

If $E$ is a non-reflexive Banach space then there exists a linear functional $\lambda \in E^*$ of norm one such that $\lambda v < 1$ for all $v \in E$ of norm one. However, in the only ...

**6**

votes

**1**answer

190 views

### Extending compact operators

Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...

**3**

votes

**1**answer

231 views

### Is $L(\ell_2,\ell_2)$ dense in $L(\ell_2,c_0)$?

Let $\ell_2:=\{ x:\mathbb{N} \to \mathbb{R}: \sum_{j=1}^\infty x_j^2<\infty\}$ and
$c_0:=\{ x:\mathbb{N} \to \mathbb{R}: \lim_{j\to\infty}x_j=0,\, \sup_{j\in\mathbb{N}}|x_j|<\infty\}$ denote the ...

**4**

votes

**2**answers

280 views

### Are Banach space norms (up to equivalence) unique?

Here is a naive question: is a "completing" norm of a vector space unique (up to equivalence) or can one find a vector space and two non-equivalent norms $\|.\|$ and $|||.|||$ that both induce a ...

**5**

votes

**0**answers

102 views

### A Banach space with all Hilbertian subspaces complemeneted

Assume that $X$ is a Banach space in which every Hilbertian subspace is complemented (let's say that all the projections are uniformly bounded). What can we say about $X$? It has to be K-convex. By ...

**0**

votes

**1**answer

130 views

### Relationship between LlogL and Hardy spaces

I think that for positive, one-dimensional, periodic functions, the following statement is true:
$$
f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}),
$$
where
$$
LlogL=\{f\in ...

**5**

votes

**0**answers

191 views

### Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space

Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...

**4**

votes

**1**answer

141 views

### Infinite dimensional subspaces of $L^1$

Suppose that $X$ is an infinite dimensional subspace of $L^{1}$. In some cases it is true that $X$ contains an isomorphic copy of an infinite dimensional Hilbert space. However, it is not the case ...

**5**

votes

**1**answer

108 views

### The Notion of Strong Measurability for Separable Banach Spaces

Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the ...

**3**

votes

**2**answers

219 views

### Generalized Hardy-Littlewood-Sobolev Inequality

The Hardy-Littlewood-Sobolev Inequality says that
$$\text{for $p,q,r\in (1,+\infty)$ such that }\quad
1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$}
$$
$$
\exists C, \forall u\in L^p(\mathbb ...

**0**

votes

**0**answers

98 views

### About the Intersection of the nested sequence of Chebyshev centers of weakly compact convex sets

Let $K_0$ be a weakly compact convex subset of a Banach space $X$. For each $n\in\mathbb{N}$, let $K_n$ be the set of Chebyshev centers of the set $K_{n-1}$. Suppose $K_0$ has a normal structure. Is ...

**3**

votes

**2**answers

179 views

### Operators on $\ell_\infty(\Gamma)$ and almost disjoint families of subsets

It is well known that given an operator $T:\ell_\infty\to\ell_\infty$ such that $Tx=0$ for each $x\in c_0$ there exists an infinite subset $M$ of the positive integers so that $Tx=0$ for each $x\in ...

**2**

votes

**1**answer

95 views

### Two basic questions on $p-$summable sequences

Let $X$ be a Banach space and $(x_{n})_{n=1}^{\infty}$ be a $p-$summable sequence in $X$. My basic questions are the following:
For any $\epsilon>0$, is there a sequence ...

**2**

votes

**1**answer

192 views

### Chebyshev centres of a bounded closed convex set in a strictly convex Banach space

Suppose $X$ is a strictly convex Banach space. Does there exist a bounded closed convex set $K$ in $X$ such that the set of all Chebyshev centers $C(K)$ of $K$ is a proper subset of $K$ with diameter ...

**9**

votes

**1**answer

438 views

### Examples of continuous differential equations with no solution

Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the Cauchy–Lipschitz theorem.
...

**0**

votes

**0**answers

82 views

### A question on $p$-approximation property

We say that a subset $K$ of a Banach space $X$ is relatively $p$-compact($1\leq p<\infty$)if there exists a $p$-summable sequence $(x_{n})_{n=1}^{\infty}$ in $X$ such that $K$ is contained in ...

**2**

votes

**0**answers

128 views

### Finitely generated groups non-embeddable into $L_1(0,1)$

I am interested in finitely generated groups which, endowed with their word metrics, do not admit bilipschitz embeddings into $L_1(0,1)$. I know two classes of such groups:
(1) Heisenberg group ...

**10**

votes

**1**answer

219 views

### Embeddings of finitely generated groups into uniformly convex Banach spaces

de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...

**1**

vote

**0**answers

75 views

### Transferring locally uniformly convex norm by bounded linear operator from one Banach space to another

I'm trying to find a simple proof this theorem
Let $Y$ be a locally uniformly convex Banach space and $X$ be a Banach space and let $T$ be a bunded linear operator from $X$ into $Y$ such that for ...

**0**

votes

**1**answer

122 views

### Two questions about convex subsets of Hilbert Space

Let H be an infinite dimensional and separable Hilbert Space. Do there exist disjoint, closed and bounded subsets A,B of H which satisfy the following conditions? (1) Each of A,B is convex and has a ...

**0**

votes

**0**answers

93 views

### Relation between modulus of smoothness and reflexivity

Baillon proved that if $X$ is a Banach space with $\rho'_X(0)<\frac{1}{2}$, then $X$ has the fixed point property (by $\rho_X(t)$
we denote the modulus of smoothness). My questions are as ...

**3**

votes

**2**answers

225 views

### Is the ideal of functions vanishing at a set complementable in $C(X)$?

Let $X$ be a compact Hausdorff topological set, and $Y$ be its closed subset. Is the ideal of functions vanishing on $Y$
$$
I=\{f\in C(X):\ \forall y\in Y\ f(y)=0\}
$$
complementable (as a closed ...

**3**

votes

**0**answers

115 views

### Decreasing sequence of closed convex sets in a Banach space

Let $(C_n)$ be a decreasing sequence of closed convex subsets in a Banach space $(E,\Vert \cdot \Vert)$. The question I have is about the content of $C=\bigcap_{n=0}^\infty C_n$.
If the $C_n$ are ...

**7**

votes

**0**answers

311 views

### The Banach space of bounded functions with countable support

Let $X$ be a set of cardinality $\aleph_1$ and consider the Banach space $\ell_\infty^c(X)$ of all scalar-valued bounded functions on $X$ which are non-zero only for countably many elements of $X$ ...

**3**

votes

**1**answer

177 views

### Positive definite quadratic forms on Banach spaces

This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it, it is called positive definite if ...

**5**

votes

**0**answers

169 views

### Large subspaces with small basic constants in finite-dimensional Banach spaces

Let $B\in(1,\infty)$. I am interested in estimates for the
function $f_B(n)$ defined as the largest $k\in\mathbb{N}$
satisfying the condition: Each $n$-dimensional Banach space
contains an ...

**0**

votes

**0**answers

49 views

### Representation of piecewise functions using variational inequalities

I need some help about how to represent piecewise smooth functions defined over N subintervals by using variational inequalities. From a paper I'm analyzing I get:
"If we can find a function ...

**2**

votes

**0**answers

52 views

### König-Wittstock norm on Banach space

Let $(E,\Vert.\Vert)$ be a real Banach space and $\ell\ne 0$
an non-continuous linear form on $E$.
Let $a\in E$ be such that $\ell(a)=1$.
König-Wittstock [Non-equivalent complete norms
and would-be ...

**1**

vote

**1**answer

89 views

### A formally weaker form of the extendable local reflexivity for Banach spaces

Rosenthal introduce the notion of the extendable local reflexivity for Banach spaces as follows: Let $X$ be a Banach space and let $\lambda\geq 1$. $X$ is said to be $\lambda$-extendably locally ...

**1**

vote

**2**answers

126 views

### The reflexivity of the space generated by a convex, balanced and compact set

Let $K$ be a convex,balanced and compact subset of a Banach space $X$. We let $X_{K}:=span\{K\}$. Define the norm $\|x\|_{K}:=\inf\{\alpha>0: x\in \alpha K\}, x\in X_{K}$. Then $(X_{K},\|\|_{K})$ ...

**8**

votes

**1**answer

222 views

### Subspaces of $l_{1}$ are not Lipschitz complemented in $l_{1}$

I have thought about the following question for several years. This question may be stupid or not interesting. My question is: Is there a subspace $U$ of $l{_1}$ such that the quotient $l_{1}/U$ is ...