All Questions
369 questions with no upvoted or accepted answers
2
votes
0
answers
199
views
Uniformly convex, uniformly smooth Banach space which is not convex of power type
It is well known that every uniformly convex Banach space $X$ admits an equivalent norm which such that it is convex of power type, i.e. the modulus of convexity with respect to the new norm satisfies ...
- 799
2
votes
0
answers
150
views
Non-separable asymptotic $\ell_1$ space
The Figiel-Johnson Tsirelson space is an example of an asymptotic $\ell_1$ Banach space not containing $\ell_1$. The notion of asymptotic $\ell_1$ is with respect to some basis, but a coordinate free ...
user114263
2
votes
0
answers
93
views
Open problems concerning Araujo's biseparating maps
Araujo stated the following four open questions at the end of his paper, page $518$ and $519.$
Question $1:$ Assume that there exists a biseparating map $T:A^n(\Omega:E)\to A^m(\Omega',F)$ which is ...
- 623
2
votes
0
answers
189
views
Dunford−Pettis property of $L^1(\mu)$
$\def\bs#1{\boldsymbol#1}\def\sp{\kern.4mm}$Let $\bs K$ be either the standard real or complex topological field, and let $E$ be a Hausdorff locally convex space over $\bs K\sp$. Then saying that $E$ ...
- 3,584
2
votes
0
answers
115
views
Mean value of a map into Banach space
Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Let $(Y,\|\cdot\|)$ be a Banach space. Given a Bochner integrable map $f:X\to Y$ with $\|f\| \in L^2(X,\mu)$. The mean value of $f$ over $X$, ...
- 169
2
votes
0
answers
143
views
About a property of bounded closed convex set
Terminology:
For a bounded closed convex (bcc for short) set $A$, define $w(A)$ to be the infimum of the distance between pairs of
parallel hyperplanes supporting $A$.
We say that a bcc ...
2
votes
0
answers
141
views
Quotients in complex interpolation of Banach spaces
Let $(X_0,X_1)$ be an admissible pair of complex Banach spaces with $X_0$ continuously embedded in $X_1$. For $0<\theta<1$, let us denote by $X_\theta =(X_0,X_1)_\theta$ the complex ...
- 4,461
2
votes
0
answers
111
views
proving that $\mathcal{A}_\infty(X)$ is or is not norm-closed in $\mathcal{L}(X)$ for each Banach space $X$
Fix any $1\leq p\leq\infty$. If $X$ is a Banach space and $C\in(0,\infty)$, we say that $T\in\mathcal{A}_C(X)$ whenever, for each $(x_n)_{n=1}^\infty\subset B_X$ (where $B_X$ is the closed unit ball ...
- 1,591
2
votes
0
answers
111
views
Ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$
I would like to know if there exist an explicit decription of ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$. The best option would be -- "they are complemented subspaces of $C(K, L_1(\mu))$ and $L_1(\...
- 1,697
2
votes
0
answers
184
views
Properties of the optimal decomposition for the $K$-functional between $\ell_1$ and $\ell_2$
Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$:
$$
\lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\...
- 1,372
2
votes
0
answers
106
views
Type-cotype inequalities for arbitrary orthonormal systems
Let $X$ be a B-convex Banach space and let $v^1 = (v^1_1,…,v^1_n), …, v^n = (v^n_1,…,v^n_n)$ be an orthonormal basis of $\mathbb{R}^n$. My question is what one can say about $\left( \sum_i \Vert \...
- 1,163
2
votes
0
answers
201
views
Reflexive subspaces of dual spaces
If $X$ is a Banach space, is it true that $X^{*}$ must contain an infinite dimensional reflexive subspace? I know of Gowers' example of a Banach space not containing $c_0$, $l_1$, or reflexive, but I ...
- 1,361
2
votes
0
answers
319
views
Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space
I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why
$d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
2
votes
0
answers
346
views
When is the sum of complemented subspaces complemented?
Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...
- 935
2
votes
0
answers
552
views
Normed space between $H^{0+}$ and $L^2$
Cosider a function $f\in L^2(\mathbb{R}^3)$ with consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty \, .\qquad\mbox{(*)} \, $$
Of course if $f\in ...
- 943
2
votes
0
answers
267
views
Banach space interpolation theory in terms of categories
I have recently learned a bit about higher category theory.
And there is a question I would like to ask. Can we enrich banach space interpolation theory by using higher category theory?
Is it ...
- 361
2
votes
0
answers
132
views
Birkhoff orthogonal of a Banach space in its bidual
Let $X$ be a Banach space embedded in $X^{**}$ in the usual way.
We consider the set
$$
O_X := \{ x^{**}\in X^{**} : \|x^{**}-x\|\geq \|x\| \textrm{ for all }x\in X\}.
$$
I think this is the ...
- 4,461
2
votes
0
answers
108
views
Suprema and infima in spaces ordered by non-normal cones
Background
We shall call a subset $V_+ \subseteq V$ of a Banach space $V$ a cone if
$V_+$ is closed,
$\alpha V_+ \subseteq V_+$ for all $\alpha \geqslant 0$, and
$V_+ \cap (-V_+) = \{0\}$.
Cones ...
- 123
2
votes
0
answers
172
views
Projective tensor powers of Banach spaces over a normed field
Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
- 371
2
votes
0
answers
648
views
Closed-form expressions for dual norms of real normed vector spaces
Didn't get any biters over at MSE, so I figure this place might be more appropriate...
Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\...
- 4,907
2
votes
0
answers
117
views
Maximum Principle with Banach Control Space
This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in which,...
- 197
2
votes
0
answers
301
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 ...
- 1,073
2
votes
0
answers
197
views
Generating cones having no surjections [in operator spaces]
Is this little toy known ?
Let $E$ be some Banach space, and let $K$ be the closed unit ball
of its dual, endowed with the weak-star topology. Also, let $j:E$ $\rightarrow$ $C(K)$
be the natural ...
- 4,060
1
vote
0
answers
87
views
Supremum of sums of functions in $L^1$ taking random signs
Consider the Banach space $X=L^1([0,1])$, and let $n\gg1$ and $x_1, ..., x_n$ be any points in the unit sphere of $X$.
Is there any reasonable lower bound for $$\sup_{(\epsilon_i)_{i=1}^n \in \{-1,+1\}...
- 393
1
vote
0
answers
105
views
Can a uniform weak null set of $c_0$ be uniformly embedded into a Hilbert space?
Remark that a uniform weak null set $A$ of $c_0$ satisfied that for any $\epsilon>0$ there exists a positive number $N(\epsilon)$ such that for every $f$ in $B(\ell_1)$, the unit ball of the ...
- 11
1
vote
0
answers
56
views
Convergence of slice in an equivalent renorming
Let us consider $\ell_2$ space with $\Vert \cdot \Vert_2$ norm. Let us define a new norm equivalent to $\Vert \cdot \Vert_2$ norm as follows:
$$
\Vert x \Vert_0 = \max \{ \Vert x \Vert_2, \sqrt{2} \...
- 85
1
vote
0
answers
80
views
Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$
This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...
- 2,830
1
vote
0
answers
73
views
Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
1
vote
0
answers
50
views
Is there $r>0$ such that the norm $[f]_r:=\sup_{n \ge 1} \|1_{B(y_n, r)} f\|_{L^p}$ is equivalent to $\|f \|:=\sup_{y \in Y}\|1_{B(y,1)}f\|_{L^p}$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(E, |\...
- 657
1
vote
0
answers
82
views
Injective envelopes of 1-extensible spaces
Please read this post as a naive follow up on a previous question.
Let $X$ be a Banach space and let $(I(X),\alpha)$ denote its injective envelope (e.g., CohenLacey1969). A low hanging fruit is the ...
- 2,605
1
vote
0
answers
47
views
Existence for a nonlinear evolution equation with a monotone operator that is not maximal
We consider the nonlinear evolution equation
$$
\dot{u}(t) + Bu(t) = 0, \quad u(0)=0
$$
with
$$
A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...
- 171
1
vote
0
answers
126
views
Non-surjective isometries of $l_p$
It is well known that all surjective isometries of $l_p$ for $p\neq 2$ are the signed permutations of the unit vector basis $(e_n)$. Is there a characterization for the linear non-surjective ...
- 1,361
1
vote
0
answers
98
views
Representations of the dual Banach algebra pair $(\ell_1,c_0)$
Let $\displaystyle E_p=(\bigoplus_{n\in\mathbb{N}} \ell^1_n)_{\ell^p}$ for some $1<p<\infty$ and $\ell^1 = \ell^1(\mathbb{N})$ be equipped with the convolution. Then, there exists an isometric &...
- 2,605
1
vote
0
answers
165
views
About a weak$^*$ convergent net
Let $G$ be a locally compact abelian group and $A$ be semisimple commutative Banach algebra such that $A^{**}$ has Radon-Nikodym property. Denote by $\Gamma$ and $M(G)$ the dual group and the measure ...
- 2,118
1
vote
0
answers
136
views
Banach spaces in which every DP-set is a limited set
Let $X$ be a Banach space and $A\subseteq X$ be a bounded subset.
$A$ is a Dunford-Pettis set if every weakly null sequence $(f_n)$ in $X^*$ converges to $0$ uniformly on $A$, that is $$ \lim_{n\to\...
- 2,605
1
vote
0
answers
151
views
Weak convergence using tensor product
I haven't got to see this argument used in the PhD thesis of [R. Ryan]: Applications of topological tensor products to infinite dimensional holomorphy, doctoral thesis, Trinity College, Dublin (1980), ...
- 203
1
vote
0
answers
71
views
Lipschitz isomorphisms of $C(\omega^\omega+)$
Let $C(\omega^\omega+)$ denote the Banach space of continuous, scalar-valued functions defined on $\omega^\omega+=[0,\omega^\omega]$. Suppose that $X$ is a Banach space and $U:C(\omega^\omega+)\to X$ ...
- 81
1
vote
0
answers
74
views
Multiple steps of the Gorelik principle
The following result is one of several non-linear Banach space theory results known as The Gorelik Principle. I am stating it here in a weaker form than what is in the literature, but the statement ...
- 81
1
vote
0
answers
61
views
When is the metric on a Fréchet space homogeneous
Let $(F,d)$ be a Fréchet space over $\mathbb{F}\in \{\mathbb{C},\mathbb{R}\}$. Are there conditions under which, there exists some $C,d>0$ such that: for every $f\in F$ and every $k\in \mathbb{F}$ ...
- 5,405
1
vote
0
answers
99
views
Module homomorphisms modulo compact operators
Let $A$ be a Banach algebra. Let $L_a,R_a:A\to A$ denote the left/right multiplication operators $$L_ax = ax, \hspace{5mm} R_ax = xa$$ for all $a,x\in A$. Assume that no nonzero $L_a$ and $R_a$ is a ...
- 2,605
1
vote
0
answers
97
views
Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
1
vote
0
answers
85
views
Interpolation between projective and injective spaces
Suppose $(\Omega,\mu)$ be a $\sigma$-finite measure space. Suppose $X$ is a Banach space and $L_p(\Omega;X)$ be the corresponding Bochner space for $0<p\leq\infty.$ Is it true that the complex ...
- 1,879
1
vote
0
answers
111
views
Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space
$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
- 2,533
1
vote
0
answers
747
views
Notation for the space of eventually-zero sequences
An eventually-zero sequence is a real-valued sequence $(x_n)_{n=1}^\infty$ for which there exists an $N\in\mathbb{N}$ such that $x_n=0$ for each $n\geq N$. The space of eventually-zero sequences ...
- 153
1
vote
0
answers
82
views
Extreme case of K-interpolation
Suppose $X_0$ and $X_1$ are Banach spaces living in a larger Banach space
$X$. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ as
$$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
- 319
1
vote
0
answers
163
views
Infinite matrices from $\ell^p$ to $\ell^{p/(p-1)}$ that are compact operators
I wanted to ask if my proof (sketch) of the following statement is correct. Namely, let $p>1$ and define $q= \frac{p}{p-1}$ we are given an operator $K : \ell^{p} \rightarrow \ell^{q}$ defined as $...
- 11
1
vote
0
answers
292
views
Closure of finite rank operators on $L^p$
It well-known that, an operator $T:H\to H$ on a Hilbert space, is compact if and only if T is limit of finite rank operators.
Besides this, the results by Per Enflo 1973 shows that this results is ...
- 1,101
1
vote
0
answers
94
views
Regularity of functions everywhere approximable by $n$-th degree polynomials
Let $(X, \lVert \cdot \rVert_X)$, $(Y, \lVert \cdot \rVert_Y)$ be two Banach spaces.
A function $P \colon X \to Y$ such that there exists $n \in \mathbb{N}$ such that for all $i \in \{ 0, \ldots, n \}$...
- 820
1
vote
0
answers
133
views
‘Linear’ intersection property of separable Banach spaces
Let $X$ be a separable Banach space. Denote $W(f,\varepsilon) = \{z\in X\colon \lvert\langle f,z\rangle\rvert < \varepsilon\}$ for some $f\in X^*$ .
Suppose that $U$ is an open set in $X$ such that ...
- 43
1
vote
0
answers
50
views
Nested nets of closed bounded star-shaped sets in a semi-reflexive space
Among Hausdorff locally convex spaces, semi-reflexivity is characterized by the weak topology having the Heine-Borel property. It follows that, in a semi-reflexive space, every nested net of closed ...
- 11