All Questions
Tagged with banach-spaces fa.functional-analysis
1,222 questions
1
vote
1
answer
151
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Is smoothness preserved under an isometric isomorphism?
Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
- 113
0
votes
0
answers
90
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How to show a point is a weak* -weak continuous for the identity map on $X_1^*$ or on $X_1^{**}$?
I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak*
-Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
- 113
5
votes
1
answer
206
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Compactness in trace class operators space
Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$.
Are there easy ...
- 59
3
votes
0
answers
158
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Gowers' dichotomy for quotients
Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable.
A ...
- 4,461
3
votes
1
answer
189
views
Ribe's Theorem: finitely representability between two uniformly homeomorphic Banach spaces
An infinite-dimensional Banach space $X$ is said to be crudely finitely representable (with constant $\lambda$) in an infinite-dimensional Banach space $Y$ if there is a constant $\lambda>1$ such ...
- 33
2
votes
0
answers
63
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Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$
In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
7
votes
2
answers
394
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Tangent space to infinite dimensional manifolds
In finite dimensional geometry, there is a single invariant of a vector space - its dimension. This characterizes finite dimensional manifolds as being glued from Euclidean balls.
This situation is ...
- 593
0
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0
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146
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On the pointwise limit of a sequence of analytic functions
I have been confused with this problem for weeks now. Suppose I have Banach spaces $E$ and $F$ and a sequence of functions $f_{n}: U \subset E \to F$, where $U$ is open and nonempty. Let $x \in U$ be ...
2
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0
answers
82
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The support of the functions in the closed span of the Rademacher functions in $L_1(0,1)$
Given a measurable function $f:(0,1)\to \mathbb{R}$, we denote by $M(f)$ the measure of the set $\{t\in (0,1) : f(t)\neq 0\}$.
It is not difficult to prove that if $(f_n)$ is a normalized sequence in $...
- 4,461
7
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1
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415
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Is there a “Closure-of-Range Theorem” for Banach spaces?
The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*$, the four conditions:
$T(X)$ is $s$-closed; $T(X)$ is $...
- 60.5k
9
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0
answers
163
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Moore-Penrose partial isometries and hermitian elements
Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
- 3,497
4
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1
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132
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Direct characterization of finite-dimensional $1$-injective Banach spaces
It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk ...
- 2,793
1
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0
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87
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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
6
votes
1
answer
335
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Existence of pairwise quasi-complementary but not complementary subspaces
Let $𝑋$ be an infinite-dimensional Banach space (complex or real). A subspace of $𝑋$ means a closed linear submanifold. Subspaces $M$ and $N$ of $X$ are quasi-complementary if $M\cap N=\{0\}$ and $M+...
- 391
3
votes
2
answers
137
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Non-complete space verifying uniform boundedness
Recently, I have seen the so-called uniform boundedness theorem, which says:
Let $(X,∥⋅∥)$
be a Banach space and $(Y,∥⋅∥)$
be a normed linear space. Let $A⊂B(X,Y)$
be a pointwise bounded family of ...
4
votes
0
answers
148
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Some questions on Hardy's spaces
In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
- 1,879
2
votes
1
answer
244
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Characterization of normed spaces based on violation of parallelogram law
For a normed linear space $(X, \|\cdot\|)$, the Jordan-von Neumann theorem specifies when exactly the norm is induced via an inner product, namely when the parallelogram law is satisfied.
I would like ...
- 213
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0
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96
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Sufficient condition for weak convergence in Banach spaces
The question is quite elementary but nonetheless no proof or counter example comes to mind immediately.
Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ ...
1
vote
2
answers
220
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A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$
Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$.
Note that $A$ is non-empty with a Baire category argument.
I ...
- 356
2
votes
0
answers
47
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Norm density of evaluation functionals in the space of weak$^*$ continuous multilinear functionals on products of dual Banach spaces
Let $K$ be a compact (metrizable) space and let $C(K)$ be the Banach space of continuous real-valued functions on $K$, equipped with the supremum norm. It is then well known that the dual space $C(K)^*...
- 56
3
votes
1
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376
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A more general product rule for weak derivatives?
Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^...
- 1,759
5
votes
2
answers
432
views
Does closedness of the image of unit sphere imply the closed range of the operator
Let $X$ and $Y$ be Banach spaces and let $T:X\to Y$ be a bounded linear operator such that $T(S_X)$ is closed in $Y$. Does it imply that $T(X)$ is closed? Any hint is appreciated.
- 585
0
votes
1
answer
96
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Existence of a complemented basic sequence
Let $X$ be an infinite-dimensional Banach space (complex or real). A subspace of $X$ means a closed linear submanifold. If $S$ is a non-empty subset of $X$, then $[S]$ denotes the closed linear span ...
- 391
7
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2
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350
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Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?
This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
- 2,830
11
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0
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342
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The diagonal operators and unconditionality
The following is well-known:
Theorem: Let $X$ be a Banach space with an unconditional basis $(e_n)_n$.
Then the space of the diagonal operators with respect the basis $(e_n)_n$ endowed with
the ...
- 986
5
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1
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220
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How big is the class of all closed range bounded linear operator?
Let $X$ and $Y$ be Banach spaces and let $CR(X,Y)$ denote the set $B(X,Y)$ of all bounded linear maps from $X$ to $Y$ with $T(X)$ closed in $Y$. Certainly $CR(X,Y)$ is not open in $B(X,Y)$ as given ...
- 585
4
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1
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158
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Is the image of a complemented subspace complemented?
This question has been crossposted from mathstackexchange:
Let $X, Y$ be two Banach spaces and $T:X\to Y$ a continuous surjection. Assume $Z$ is a complemented subspace of $X$ and that $T(Z)$ is ...
- 655
3
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0
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165
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$S^{n}(V)$ is "approximately" $V$ when $n$ goes to infinity (in the setting of normed space)
Let $B$ be a (separable) Banach space. $(v_{i})_{i}, i\in\mathbb{N}$ being a family of linear independent vectors. $V$ being the span of $v_{i}$. I try to prove that $V$ is dense in $B$. I define a ...
- 523
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0
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78
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What does analytic uniformly in $s$ mean?
Suppose I have a complex vector space $V$ with finite basis $\{e_{1},...,e_{s}\}$. Then, I can consider the algebra $\mathcal{U}$ of formal polynomials on the variables $e_{1},...,e_{s}$. Suppose ...
3
votes
1
answer
207
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Embedding of a Banach space in $\ell_\infty(\Gamma)$ with a subspace embedded in a copy of $\ell_\infty$
In this question (which I think may be interesting in its own) I was asking if we can find a copy of $\ell_\infty$ between a separable subspace $Y$ contained in $\ell_\infty(\Gamma)$ and the whole ...
- 271
1
vote
1
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209
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Rate of convergence of mollified functions in $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
- 835
3
votes
1
answer
169
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Copy of $\ell_\infty$ inside $\ell_\infty(\Gamma)$ containing given subspace
To complete a proof I need to know if the following is true:
Given a non-empty set $\Gamma$ and a separable subspace $Y$ of $\ell_\infty(\Gamma)$, there exists a subspace $A$ of $\ell_\infty(\Gamma)$ ...
- 271
6
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1
answer
290
views
Analytic maps on Banach spaces: analyticity upgrade
Consider the following problem.
Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and
$$ f:U\to G $$
an analytic map, such ...
- 1,075
2
votes
0
answers
82
views
What is Lipschitz constant of the radial renormalization $(X,\|\cdot\|_a) \rightarrow (X,\|\cdot\|_b)$ on a normed vector space $X$
Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping
$$
f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X,
$$
with $f(0)=0$
is a radial and maps ...
- 5,327
0
votes
1
answer
151
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Super-reflexivity is separately determined
I've found this result that states that super-reflexivity is separably determined, i.e., if every separable subspace $Y\subset X$ of a Banach space is superreflexive then $X$ itself is superreflexive. ...
- 145
2
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1
answer
148
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Is projection of a closed subspace Borel?
Specifically, letting $E$ be a separable infinite-dimensional real Banach space, and $D_2$ in $E\times E$ a closed linear subspace, is then $\{\,x:\exists\,y\,;(x,y)\in D_2\}$ a Borel set in $E\,$? ...
- 3,584
2
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0
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96
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Isometric Schröder-Bernstein theorem for injective Banach spaces?
It's known that every injective Banach space is of the form $C(M)$ where $M$ is a compact, Hausdorff, extremally disconnected topological space.
Let $X$, $Y$ be two injective Banach spaces such that,
...
- 2,605
0
votes
1
answer
106
views
Convergence of mollified functions in weighted $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
- 835
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
7
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2
answers
345
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Integral means vs infinite convex combinations
Let $(X,\mathcal A, \mu)$ be a probability space, $\mathbb E$ a Banach space, and $f:X\to\mathbb E$ a Bochner integrable function.
Does there exist a sequence $(x_k)_{k\ge 1} $ in $X$, and a ...
- 60.5k
2
votes
1
answer
136
views
Is there a scalar product which makes orthonormal the family of complex functions $ (f_n)_{ n \geq 1 } $?
Let $ (f_n)_{ n \geq 1 } $ be a family of complex functions defined as follow,
$ \forall n \geq 1 $,
$$ f_n (z) = \dfrac{1}{n^{z}} $$
I would like to ask you if it is possible to construct a ( non-...
- 595
7
votes
3
answers
909
views
Using the Stone-Weierstrass theorem to solve an integral limit
The following question was posted on math stack exchange here but it got no answers
Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
- 331
2
votes
0
answers
184
views
Example of space which is weak Hahn-Banach smooth but not Hahn-Banach smooth
A Banach space $X$ is said to be Hahn-Banach smooth if every linear functional on $X$ has a unique norm-preserving extension over $X^{**}$. Weak Hahn-Banach smoothness is what if the above condition ...
- 521
2
votes
0
answers
88
views
Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
- 21
5
votes
1
answer
561
views
interiors of positive cones in ordered Banach spaces
I have a couple of questions about ordered Banach spaces and interiors of their positive cones. I would appreciate your insights and any recommended references.
I want to know several examples of ...
- 79
0
votes
1
answer
317
views
A variation of the Riesz Lemma
Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
- 153
-1
votes
1
answer
139
views
$L^1$ convergence
Setting
For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
- 125
0
votes
1
answer
163
views
Counterexample wanted: Banach space but not BK-space
What is an example of a Banach space that is not a BK-space?
A normed sequence space $X$ (with projections $p_n$) is a BK Space if $X$ is Banach space and for all natural numbers $n$, $p_n(\bar{x}) = ...
- 1
4
votes
1
answer
165
views
Dual spaces of Banach-valued $L^{p}$-spaces
Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\...
- 1,429
0
votes
0
answers
49
views
Kadec-Klee property of an equivalent norm on a Hilbert space
Let us consider the space $\ell_2$ with the Hilbert norm $\Vert \cdot \Vert$ and consider the following eqivalent norm:
$$
\Vert (r,x) \Vert_A^2 = \Vert (r, Tx)\Vert^2 + \max \{ \Vert x \Vert, \vert r ...
- 85