# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

8,108
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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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62
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### Cousin of Fourier transform for rescaling and translating functions

Is there a cousin of the Fourier transform which obeys the following property: if I rescale and translate a function, then its transformed form, at each frequency, only depends on the original ...

1
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1
answer

132
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### Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$?

Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all ...

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1
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### Applications and motivations of resolvent for elliptic operator

Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is
\begin{align}
\mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2
\...

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### Norm inequalities for sums of two basic elementary operators

I'm working on this paper. What I'm interested in is this theorem:
where $M_{A,B}(X)=AXB$
I don't know why we can find those two sequences $X_n$ and $x_n$, either I'm finding difficulties to show the ...

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0
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70
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### What's the problem with the evaluation map not being continuous?

When introducing differentiable functions between locally convex spaces, many authors (e.g. Bastiani, Keller, Kriegl-Michor) notice that the evaluation map
$$ E\times E^*\to\mathbb R,\qquad (x,L)\...

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135
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### Explanation of a step in a preprinted work

I have been studying this preprinted paper and the references therein. I believe that there are some typos; However, since I am not an expert yet, I would like to make sure I am correct.
I do not ...

3
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### Generalized Jensen's inequality for positively homogeneous functions

The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...

3
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1
answer

108
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### Urysohn's lemma for Bochner functions?

Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used:
If $U$ is an open ...

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0
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36
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### Some simple conditions on a function $f$ so that $x\mapsto xf(x)$ is convex?

Studying the Braess Paradox for a project at school (with the assiociated Wardrop's equilibria and Nash's game theory result) I came upon one simple question I can not figure out….
If $f$ is a ...

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1
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26
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### Closure of the point spectrum of an unbounded diagonalizable operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...

2
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33
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### duality argument

Throughout my studying for some papers, in particular, the proof of localized Strichart estimates, I encountered the use of the duality argument I could not fully understand. The outline of the ...

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42
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### What is lost after RKHS embedding of the L1 space?

We know that every distribution or $L^1$ function $f$ over space $\mathcal{X}$ (e.g., $R^d$) can be embedded to an RKHS $\mathcal{H}$ with a $1$-bounded kernel $\mathcal{K}$ (e.g., the RBF kernel) ...

2
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1
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91
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### Convex series and closed convex hulls in normed spaces

Let $(X, \lVert \cdot \rVert)$ be a normed space over $\mathbb{R}$ and $A = \{ a_1,a_2 \ldots \} \subseteq X$ be a closed bounded set.
Let $\overline{\mathrm{co}}(A)$ denote the closed convex hull of ...

3
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101
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### Examples of infinite dimensional involutions

Examples of infinite dimensional involutions
I'm looking for more examples of involutions of the type portrayed below, in which two sets of indeterminates (real or complex) each can be transformed ...

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46
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### Relation between the norm of Sobolev space $H^1$ and $L^p$ norm for non-increasing radial functions

I am interested to find $$\sup\|u\|_{p}^{p},$$ when $u$ are non-increasing radial functions on the unit ball $B_1$ of $\mathbb{R}^{n}$ such that $$\|u\|_{H1}^2 < r$$ for some $r > 0$.
Since $u$ ...

4
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1
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198
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### Takesaki lemma 1.16 (volume II, chapter VII)

I am trying to understand the proof of the implication $(i)\implies (ii)$ in Takesaki's book "Theory of operator algebras II", chapter VII, which says the following:
The relevant setting ...

2
votes

1
answer

71
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### Questions about ratio set in a dynamical system

Given a dynamical system $(X, \Omega, \mu)$ ($\Omega$ the $\sigma$-algebra and $\mu$ a measure), we assume there is a group $G$ acting on the system in the sense that, for each group element $g$, $g$ ...

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1
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65
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### References and updates on a $L^p$ Factorization theorem by Maurey

In the "Exposé XV" of the 1972-1973 of the Maurey-Schwartz Seminar of functional analysis ("Théorèmes de factorisation pour les opérateurs linéaires à
valeurs dans un espace $L^p(\Omega,...

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0
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77
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### Is the integral functional analytic?

Let $E = C^k(\overline{U})$, $k \geq 1$, be the Banach space of real functions of class $C^k$ on a bounded open set $U \subset \mathbb{R}^n$, with its usual norm. Let $I : E \to \mathbb{R}$ be the ...

3
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1
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151
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### Explanation of a step in a work by C. E. Kenig and A.D. Ionescu

I am studying the work
Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...

2
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0
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49
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### The Laplace transform and the Lagrange compositional inversion formula

I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...

0
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1
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52
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### Strong measurability of operator-valued map induced by a kernel

Let $K\in L^2([0,T]^2)$, and for each $t\in [0,T]$, let $\mathcal{T}_t $ be such that for all $f\in L^2([0,T])$, $\mathcal{T}_t f(s)=\int_0^T K(s,t)K(u,t)f(u)\, d u$ for all $s\in [0,T]$. One can show ...

3
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133
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### $C^1$-regularity of solution of a Dirichlet problem

I've been stuck trying to hunt down a proof/reference for a certain regularity result which I need for my thesis (all the authors I've consulted refer to it as "well-known"). Consider the ...

4
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1
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114
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### Spectral theory of infinite volume hyperbolic manifolds

I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...

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3
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200
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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, \...

5
votes

1
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203
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### de Rham theorem for tempered distributions

I am wondering if the following statement holds.
If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...

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79
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### Self-ajointness of the Laplacian over a Riemannian manifold with boundary

I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf).
Let
$(M,g)$ be a Riemannian manifold with boundary;
$E\to M$ be an hermitian fiber bundle;
$\Delta$ ...

2
votes

1
answer

338
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### Positiveness of Banach limit [closed]

I‘m currently reading Arveson’s “A Short Course on Spectral Theory”, and I’m stuck at Exercise 3.1 (1). The question is:
Let $l^{\infty}(\mathbb{N})$ be the set of all bounded sequences of complex ...

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99
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+50

### Prove comparison principle for $u_t + f(u)_x = g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$

Let us consider
$$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f \in W^{1,\infty}(\mathbb R)$ and $f \not \equiv 0$ (possibly, we can also add the ...

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98
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### Time evolution in finite-dimensional quantum mechanics [closed]

This question arose more like a curiosity and maybe more suitable for mathstack, but I posted the question there and got no answers, so I am posting it here as well.
Let $M$ be a linear operator on a ...

5
votes

1
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125
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### Describing the Gamma-transform explicitly in terms of power series

The Gamma transform of a measure is defined as follows. If $\alpha$ is a $\mathbf{Z}_p$-valued measure on $\mathbf{Z}_p$, then the Gamma transform of $\alpha$ is:
$$\Gamma_{\alpha}(s) = \int_{\mathbf{...

4
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2
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94
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### Is a unital $*$-morphism from a unital $C^*$-algebra $A$ to $\operatorname{End}_{\mathbb{C}}(K)$ automatically contractive?

Let $A$ be a unital $C^*$-algebra and let $K$ be an inner product space (not necessarily complete!). Let $\pi: A \to \operatorname{End}_{\mathbb{C}}(K)$ be a unital algebra homomorphism such that
$$\...

1
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1
answer

75
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### Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...

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31
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### Classic Hölder spaces versus Hölder-Zygmund spaces on Riemannian Manifolds

Let $M$ be a compact Riemannian manifold without boundary.
The spaces ${C}^k(M)$ are defined as usual for $k \in \mathbb{N}$ and we can define Hölder spaces ${C}^s(M)$ $s \geq 0$, $s \notin \mathbb{N}$...

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0
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34
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### Action of fractional Laplacian on Hölder / Besov spaces on Riemannian manifolds

Let $M$ be a compact Riemannian manifold (without boundary) and $\Delta$ be the corresponding (positive) Laplace-Beltrami operator. We also define the operators $I^s = (\mathrm{Id} + \Delta)^{-s}$ for ...

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0
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46
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### Sobolev variant of Wasserstein space

Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...

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### Regularity of Volterra convolution integral

I have a question about the regularity of the convolution integral. Let $f: [0,\infty) \to \mathcal{L}(\mathbb{R}^n)$ given by $f(t) = e^{tA}$. Let $g: (0,T) \to \mathbb{R}^n$ such that $g \in L^2(0,T;...

1
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0
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92
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### $C^0$ norm is bounded by $L^{14}$ norm

Let $M$ be a closed manifold of dimension $6$, and we look at the collection of smooth functions on $M$ which satisfy:
\begin{align*}
||f||_{C^0}\leq C\big(||f||_{L^{14}}^2+1\big)
\end{align*}
for ...

7
votes

1
answer

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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 $T\otimes S\colon X \otimes_\pi W\to Y \otimes_\pi V$ ...

2
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### Smoothness of Radon transform

Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...

0
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1
answer

83
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### How to calculate the dual spaces of the following spaces?

Let us consider the spaces $C_\infty(E)$, $C_c(E)$, $C_b(E)$, where $E$ is locally compact, $C_\infty(E）$ is all continuous functions vanishing at the ends of $E$, $C_c(E)$ is all the continuous ...

3
votes

1
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86
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### Estimate for an oscillatory integral of the first kind

I am confused in finding the right bound for the following oscillatory integral
$$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$
Where $\psi(2^{-k} \xi)$ is a smooth ...

1
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0
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59
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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 ...

2
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0
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### $f\in L^2_k(\mathbb R\times \mathbb S^1)$ implies that $t\mapsto f(t) \in L^2_a(\mathbb R, L^2_{k-a}(\mathbb S^1))$?

$\newcommand{\SS}{\mathbb{S}^1}$
$\newcommand{\R}{\mathbb{R}}$
Consider a function $f:\R\times \SS\to \R$ and suppose that $f$ is in the Sobolev space $L^2_k(\R\times\SS)$ for $k>1$ so that we can ...

3
votes

1
answer

96
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### Impact of annihilators in C*-algebras

Let $A$ be a unital C*-algebra. Let $S\subseteq A$. We put $$\operatorname{Ann}_r(S)=\{a\in A : \forall s\in S,~ ~as=0\}$$
Suppose that $A$ satisfies the following property:
For every $S\subseteq ...

5
votes

0
answers

76
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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 ...

0
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0
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60
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### Converse to Cameron-Martin theorem

It is known by Cameron-Theorem that if $\mu$ is a centered Gaussian measure on Banach space $\mathcal B$, the equivalent mean-shift measures are exactly the mean-shift by the Cameron-Martin directions....

4
votes

1
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165
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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 } \...

3
votes

1
answer

125
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### Does property (V) imply the Grothendieck property for dual Banach spaces?

A Banach space $X$ has property (V) whenever for each Banach space $Y$, every unconditionally converging operator $T:X\to Y$ is weakly compact; equivalently, every non-weakly compact operator $T:X\to ...