# Questions tagged [banach-spaces]

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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### The embedding of a Banach lattice in an ultrapower

Given a Banach space $X$ and a non-trivial ultrafilter $\mathcal{U}$ on a set $I$, the ultrapower $X_\mathcal{U}$ is defined as the quotient of $\ell_\infty(I,X)$ by the closed subspace $N_\mathcal{U}(...

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### Is $\phi(t)=\|P(w+td)-w\|_X/t$ nonincreasing if $X$ is "only" a uniformly smooth and uniformly convex reflexive Banach space?

For a Hilbert space $X$ it is known that the function $\phi(t)=\frac{1}{t}\|P(w+td)-w\|_X$ with $t>0$ is nonincreasing. Here, $P:X\to C$ denotes the projection operator and $w \in C, d \in X$ are ...

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### Studying an article - Sobolev inequality

Good morning people. I am not able to understand the passage (which I will address next) in the following article
https://sci-hub.se/10.1002/mana.201200319
Let us define
$$
\mathbb{H}^{1} = H^{1}(-L,...

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### Example of a compact operator that is not uniformly continuous

I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ...

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### The complement of $L_1(0,1)$ in $L_1(0,1)^{**}$

Let $\mu$ be a finite measure, like the Lebesgue measure in $(0,1)$. It is well-known that $L_1(\mu)$ and its second dual $L_1(\mu)^{**}$ are Banach lattices, $L_1(\mu)$ is a projection band in $L_1(\...

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### Is there a reflexive Banach space whose ball is not the convex hull of its extreme points?

Let $X$ be a reflexive Banach space. Then the convex hull of the extreme points of the unit ball is weakly dense by the Krein-Milman theorem and Kakutani's theorem. My question is, if there is an ...

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### Isomorphic embedding of $l^n_{\infty}$ into $l_1^m$?

Given $n$, is there a $C(n)$-isomorphic embedding of $l^n_{\infty}$ into $l^m_1$ for sufficiently large $m$ and $C(n)<<\log(n)$?
For $n=2$ this can done with $m=2$. There are some results about $...

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### Get an estimate on $L^{2}(0,1)$

Consider $f \in L^{2}(0,1)$ and $g \in L^{\infty}(0,1)$ such that
$ \text{lim} ~g(x) = 0 \ \ \text{when} \ \ x \to 0^{+};$
$g(x) > 0 \ \forall x \in (0,1)$;
$\text{lim}~\dfrac{g(x)}{x^{\alpha}} =...

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### 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 ...

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### Properties on morphism of locally convex vector spaces

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $U,V,W,W'$ $K$-vector spaces, such that $U$ is a Banach-space and $W,W'$ are finite dimensional. Further we have an (algebraic) short exact ...

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### Contact points for John's ellipsoid

Suppose $K$ is a centrally symmetric convex body in $\mathbb{R}^n$ and $E$ is the John's ellipsoid, the ellipsoid of maximal volume inside $K$.
If $E$ and $K$ have exactly $2n$ contact points, say $(\...

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### A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$

Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $...

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### About the constructions of Tsirelson space

Let $T'$ be the Tsirelson space dual in the Figiel-Johson construction, and $B_{T'}$ the unit closed ball. Then, $B_{T'}$ satisfies the next properties:
(i) $B_{T'}\subset B_{l^\infty}.$ Each vector ...

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### Morphism in commutative square strict?

Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism.
Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...

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### A reproducing kernel Hilbert space

A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e....

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### Parallelogram law for vectors of equal length

Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is,
if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert ...

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### A space isometric to $\ell_\infty^2$

Consider a norm on $\mathbb C^2$ as $\|(z_1,z_2)\|:=\max\{|z_1|,|z_2|,\frac{1}{\sqrt{2}}|z_1+iz_2|\}.$
Question. Is $(\mathbb C^2,\|.\|)$ linearly isometric to $(\mathbb C^2,\|.\|_{\infty})$ where $\|(...

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### Subspaces of $\ell_\infty^3$

Let $a,b\in\mathbb C$ be suc that $\max\{|a+b|,|a-b|\}\leq 1$ but $|a|+|b|>1.$ According to this paper by Arias, Figiel, Johnson and Schechtman https://www.jstor.org/stable/2155206?origin=crossref#...

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### Isometric embeddings of $c_0$ into metric spaces

Are there any nice and useful criteria or theorems which assert when a given metric space $M$ contains an isometric (not necessarily linear) copy of the Banach space $c_0$ or its unit ball $B_{c_0}$? (...

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### when a quasinilpotent is nilpotent

In the case of an infinite-dimensional complex banach space X,
under what conditions can a quasinilpotent operator T\in B(X) be determined to be nilpotent?

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### The solid hull and weak*-closure in Banach lattices

Let $E$ be a Banach lattice. A subset $A$ of $E$ is called solid if $|x|\leq |y|$ for some $y\in A$ implies that $x\in A$. For a subset $A$ of $E$, the solid hull of $A$ is the smallest solid set ...

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### Determine if an integral expression is in $L^2(\mathbb{R})$

Note: This is a simplified version of the following question. I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a ...

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### Compactness of the unit ball of a Banach space for topologies finer than the weak* topology

Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\...

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### Is Baire's theorem stronger than needed for functional analysis?

Many classic theorems in functional analysis involve using Baire's theorem to prove facts about topology that relate to maps between Banach spaces (or, more generally, F-spaces). The application ...

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### Fractional integration in Orlicz spaces

I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley.
And I would like to understand one question:
Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...

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### Boolean algebras generated by order intervals of projectional skeletons

A projectional skeleton (PS) on a Banach space $X$ is a family of projections $\mathcal{P}=\{P_i : i\in J\}$, indexed by a directed and $\sigma$-complete set $(J,\leq)$, satisfying
$P_i(X)$ is ...

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### 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 $k\in \mathbb{F}$ ...

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### Extracting block sequences from $\ell_p^n$-like sequences in isomorphic copies of $\ell_p$

One of the characterizing properties of the canonical bases of the $\ell_p/c_0$ spaces is their perfect homogeneity (Zippin), so that every normalized block sequence in these spaces behaves like the ...

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### Maximum likelihood estimation for covariance parameters of infinite dimensional Gaussian measures

Assume that $\mu_\theta$ is an infinite dimensional Gaussian measure on a Banach or Hilbert space with a covariance operator $C_\theta$ where $\theta$ denotes a finite dimensional vector of real-...

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### Dense subset of Hardy space of semi-martingales

Fix $p\in [1,2]$, a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$, and let $\mathcal{H}_{\mathscr{S}}^p$ denote the space of semimartingales $X$ such that the norm
$$
\...

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### 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 ...

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### Quotients of $c_0$ that are complemented in $c_0$

Suppose $X$ is a closed subspace of $c_0$ with an unconditional basis and suppose also that it is a quotient of $c_0$. Is $X$ also a complemented subspace of $c_0$?
An affirmative answer implies that $...

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### Finite dimensional subspaces of L^p, entropy estimates

The following follows from Proposition 9.6 in Approximation of zonoids by zonotopes, Acta Math. 162 (1989), no. 1-2, 73–141, by Bourgain, J., Lindenstrauss, J., and Milman, V.
Let $X\subset L^1(\mu)$ ...

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### 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). ...

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### Example when Lorentz-Shimogaki condition satisfied with a specific Young's function

Let $\Psi(t)=\int_0^t\psi(s)$ be a Young's function. Then, $\Psi$ satisfies the Lorentz-Shimogaki condition if
$$
\int_0^{\infty}\frac{\Psi(st)}{v(t)^2}\psi(t)dt< \infty.
$$
Denote $\rho_{\Psi}=\...

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### Example of the bounded convolution operator when Sharpley's conditions does not hold

I am reading about Orlicz and Marcinkevich spaces, and wondering whether there is an example in which Sharpley's condition is not satisfied for a special bounded operator $T_k$ (see for reference ...

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### Weak*-null sequences in dual spaces

Let $(x_{n})_{n}$ be a sequence in a Banach space $X$. Assume that the set $\{x_{n}:n=1,2,\cdots\}$ is finite. Let $(f_{m})_{m}$ be a weak*-null sequence in $X^{*}$ satisfying the following conditions:...

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### Weakly null sequences in Banach spaces

Every weakly null sequence in a Banach space, as a subset, is clearly relatively weakly compact. To quantify the elementary fact, we need the following quantities:
$$\delta_{0}((x_{n})_{n}):=\sup_{x^{*...

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### Boundedly complete bases

Let us recall that a basis $(x_{n})_{n}$ for a Banach space $X$ is boundedly complete if for every scalar sequence $(a_{n})_{n}$ with $\sup\limits_{n}\|\sum\limits_{i=1}^{n}a_{i}x_{i}\|<\infty$, ...

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### Regarding significance of spectral variation under algebraic operations

I have been reading the paper Determining elements in $C^∗$-algebras through spectral properties.
The paper discusses about what would be the relation be between two elements $a$ and $b$ of a Banach ...

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### 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 ...

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### Why is it difficult to define a direct integral of Banach spaces or Banach algebras?

In the relevant Wikipedia entry, I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras.
Suppose that I want to define a direct integral on either Banach spaces or ...

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### Complex Hölder space

I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...

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### What will be the upper bound of covering number of $r B_{n}^2$?

I am very curious to know the covering numbers of the $r$-radius Euclidean ball $B_{n}^2$ for any $r > 0$: so
$r B_{n}^2$ is a Euclidean ball with radius $r$.
I have read Covering number of $l_2$ ...

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### Embedding finite dimensional subspaces of Schatten p-classes

I have recently read a paper by Talagrand: Embedding subspaces of $L_{1}$ into $l^{N}_{1}$, and part of the chapter: Finite dimensional subspaces of $L_{p}$, written by Jonson and Schechtman. The ...

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### Squeezing more convergence from the convergence in all $L^p$ spaces

Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...

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### Projective tensor product of injective operators

I've seen claims that it is known that for a pair of bounded injective linear operators $T\colon X\to Y, S\colon W\to V$, their tensor product $T\otimes S\colon X \otimes_\pi W\to Y \otimes_\pi V$ ...

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### 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 ...

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### Is the Grassmannian of a Banach space an infinite dimensional manifold?

Grassmannian of complemented subspaces in a Banach space is a Banach manifold. This is explained for example in the thesis of Douady and is rather analogous to the finite-dimensional case.
I would ...

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### ODE in Banach space

Have I understood this correctly:
So originally we consider the following partial differential equation:
$$u_t= \frac{u_{xx}}{1+w}-\frac{1}{\epsilon}(1+w)u^3+\frac{wu}{\epsilon(1+w)} \text{ in } \...