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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Hankel Operator is compact
I am currently working on compact operators. I am trying to solve the following exercise
Problem
Let $(a_j)_{j \in \mathbb{N}}$ be a sequence of complex numbers in $\ell_1$, i.e. $\sum_j |a_j| < \...
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Strong convergence of compact operator
I am currently doing an exercise on compact operators.
Problem
Let $K: X \to Y$ be a compact linear operator. Suppose $(x_{n})_{n \in \mathbb{N}}$ is a sequence in $X$ with the property that there ...
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Space of functions of infinitely many variables with norm different from the uniform one
Consider a family of compact metric spaces $(\Omega_t)_{t\in\mathbb{R}}$, and the corresponding product space $\Omega=\prod_{t\in\mathbb{R}}\Omega_t$. I will write its generic element as $\omega=(\...
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Finding the conjugate of an operator between the Banach spaces $\ell_{p}$
I am working with conjugate operators acting between Banach spaces. I am doing the following exercise.
Let $(\beta_{n})_{n \in \mathbb{N}}$ be a bounded sequence of complex numbers. Define the ...
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Is the duality product of dual spaces unique?
Let $\Omega \subset \mathbb{R}^n$. Consider, for example, the Sobolev space $H^1_0(\Omega)$. It is known that the dual is $H^{-1}(\Omega)$ and is Banach with respect to the operator norm
$$||f||_{-1} =...
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Every metric space can be embedded isometrically into the Banach space
I've been trying to construct my own proof of this, but the only reference I've been able to find that I could (somewhat) understand was the following. In particular, I'm confused at the use of $f_{a}(...
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A possible characterization of WSC spaces
A Banach space $X$ is weakly sequentially complete (WSC) if every weakly Cauchy sequence in $X$ is weakly convergent.
I will use the following classical result:
Rosenthal's $\ell_1$ theorem: Every ...
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Do closed, linear spaces always contain non-empty open sets?
For regular spaces (such as Banach spaces) we know that non-empty, open sets contain closed subsets.
The reverse shouldn't always be true (e.g. if we take singletons). Now, I have a non-trivial, ...
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Test the sequence of functions $x_n={e^{-nt}}$ for convergence in $C[0,1]$,$L_{1}(0,1)$ and $L_{2}(0,1)$.
Test the sequence of functions $$x_n(t)={e^{-nt}}$$ for convergence in
$C[0,1]$,$L_{1}(0,1)$ and $L_{2}(0,1)$. In case of convergence, find
the limit function.
What I have done.
(1) In $C[0,1]$
For $...
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"Marginal" distributions of probability measure on separable Banach space
Let $(\Omega,\mathcal F,P)$ be a probability space, and consider a random variable $X: (\Omega, \mathcal F)\longrightarrow (S,\mathcal S)$, where $S$ is a separable real Banach space. It is well-known ...
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Representation of $L_2$ norm with duality, in a Banach space [closed]
Let $(E,\|\cdot\|,\mathcal{F})$ be a Banach probability space, let $\phi:E\rightarrow\mathbb{R}$ be a lower-
semicontinuous convex function, and let $\phi^*$ be its convex conjugate. For any $x\in E$,...
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Continuity of Linear Map on Tensor Product Spaces with Different Norm Properties
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and extending this to a map $\phi^k: V^{\otimes k} \rightarrow U^{\otimes k}$ defined by
$$
\phi^k(v_1 \otimes \...
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Weak limit of a bilinear operator
I have a bounded bilinear operator $B:X\times Y \to Z$, where $X,Y$ and $Z$ are Banach spaces. That is linear in both variables and satisfies:
$$\|B(x,y)\|_Z \le \|B\|\ \|x\|_X\ \|y\|_Y, \quad \forall ...
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Koranyi norm in Heisenberg group gives a Banach space structure?
The $(2N +1)-$dimensional Heisenberg group $\mathbb{H}^N$ is the space $\mathbb{R}^{2N+1} = \left\{ (x,y, \tau ) \right\}\in \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}$ equipped with the ...
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$C'[0,1]$ is a Banach space with the norm $||f||=\max_{t\in[0,1]} \{|f(t)|,|f'(t)|\}$
Let $C'[0,1]$ be the space of real functions defined in $[0,1]$
continuously differentiable in $(0,1)$ which derivative can be extended continuously to $[0,1]$. Show this is a Banach space with the ...
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Show that $\|T^n\|=\frac{1}{n!}$ for $(Tf)(x)=\int_0^xf(t)dt, f\in C[0,1]$
Let $C[0,1]$ and define,
$$ (T f )(x) =\int_0^x f (t) dt, \ x ∈ [0, 1]$$
for $f ∈ C[0, 1]$. Show that $\|T^n\|=\frac{1}{n!}$
By definition, I know that $\|T\|=\sup_{\|f\|=1}\|Tf(x)\|$, using the norm ...
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what exactly is the defintion of Y-weak topology?
Recently, I read a textbook by Barry Simon,and it is opertor theory, a comprehensive course in analysis part 4. In the Banach space notation part, he says that: For $X$ be banach space, $\sigma(X,Y)$=$...
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Power series, but in p-norms
This might seem a bit left-field, but it came up rather naturally in some work I've been doing recently in machine learning and I can't seem to track down any work along the same lines. In essence, ...
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Why do we use this norm on sequence spaces
I'm studying sequence spaces $\ell^p=\{(x_j)_{j\in \mathbf{N}}:\sum_{j\in \mathbf{N}}|x_j|^p<\infty\}$ for $1\leq p<\infty$.
This is a vector space (I'm not sure how to prove it is closed under ...
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Boundedness of Linear Operators on Banach Subspace with Different Norm
I had this exercise on a functional analysis exam but I was unable to solve point iii).
I solved points i), ii), and iv). I solved i) with the open map theorem, ii) using i) and iv) as an instance of ...
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Example of isometry between reflexive and non-reflexive Banach spaces
Let $E,F$ be real Banach spaces where $E$ is reflexive. Let $i:E \to F$ be a surjective isometry. If $i$ is linear, then $F$ is also reflexive.
Could you give an example where $F$ is not reflexive?
...
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Projective tensor product of operator spaces
Consider the following fragment from Effros and Ruan's book "Operator spaces"
Why is a decomposition as in the red box possible? In fact, it is not even clear to me that any such ...
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Abstract index is an invariant of Banach algebras
I just started to read the book An introduction to Operator Algebras by Kehe Zhu. In this book the author defines the index group of a Banach algebra $A$ (which is assumed to be a unital algebra) as ...
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Is there any normed space which does satisfies paralleogram law but not a Hilbert space?
Parallelogram law is a necessary condition for a Banach space to be Hilbert, but it is not sufficient.
Can anyone give an example of that kind of normed space, which satisfies
$$
\|x+y\|^2+\|x-y\|^2=2\...
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Let $\mathcal{B}_0(X, Y)$ is a Banach space. Does this imply that $Y$ is a Banach space?
Consider $$\mathcal{B}_0(X, Y) =\\{T\in\mathcal{B}(X,Y): \overline{T(B_X[0,1])}\subset Y \text{compact}\\}$$
where $B_X[0, 1]=\{x\in X: \|x\|\le 1\}$
Claim: $\mathcal{B}_0(X, Y)$ is a Banach space ( ...
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Separation of normalized molecules in Lipschitz free space
For a pointed metric space $(M,d,0)$ (meaning $(M,d)$ is a metric space and $0\in M$ is a distinguished point), $\text{Lip}_0(M)$ is the Banach space of all real-valued Lipschitz functions $f:M\to \...
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Does Amann's Theorem 1.4 about $\mu$-measurability extend to metrizable topological groups?
Let $(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space and $(E, | \cdot |)$ a Banach space. Let $f:X \to E$. We recall some definitions at page 62 of Amann's Analysis III.
$f$ is ...
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Frechet differentiability, RNP, and Asplundness (soft question)
I'm looking for references which provide good surveys and histories of results related to Gateaux and Frechet derivatives and their relationship to Asplundness and RNP. Specifically, any papers or ...
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Hausdorff separation for the definition of Mackey topology
I am reading the definition of the Mackey topology relative to a dual system $(X,Y)$ and the author (Edwards-Functional Analysis) imposes the condition that the dual system must be separated in $Y$. ...
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If $f \in L^0 (X, L^p_{\text{loc}} (Y))$, then $f \in L^0 (Z)$
Below we use Bochner measurability and Bochner integral. Let
$T>0$ and $p \in [1, \infty)$,
$X :=[0, T]$ and $Y:= \mathbb R^d$,
$\cal A$ the Lebesgue $\sigma$-algebra of $X$,
$\cal B$ the Lebesgue ...
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If for a.e. $x \in X$ the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^p_{\text{loc}} (Y)$, then $(f_n)$ is Cauchy in $(L^0 (Z), \rho_Z)$
Let
$T>0$ and $p \in [1, \infty)$,
$X :=[0, T]$ and $Y:= \mathbb R^d$,
$\cal A$ the Lebesgue $\sigma$-algebra of $X$,
$\cal B$ the Lebesgue $\sigma$-algebra of $Y$,
$\mu, \nu$ complete finite ...
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Convergence in marginal measure implies that in product measure
Let
$T>0$ and $p \in [1, \infty)$,
$X :=[0, T]$ and $Y:= \mathbb R^d$,
$\cal A$ the Lebesgue $\sigma$-algebra of $X$,
$\cal B$ the Lebesgue $\sigma$-algebra of $Y$,
$\mu, \nu$ complete finite ...
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Let $\|f_n-f\|_{L^p_{\text{loc}}} \to 0$. There is a subsequence $(n_k)$ such that $f_{n_k} \xrightarrow{k \to \infty} f$ a.e.
Let $p \in [1, \infty)$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $f:Y \to \mathbb R$ such that
$$
\|f\|_{L^p_{\text{loc}}} := \sup_{y \in Y} \|1_{B(y, 1)} ...
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Assume $f(x, \cdot) \in L^p_{\text{loc}} (Y)$ for a.e. $x \in X$. Then the map $x \mapsto \|f(x, \cdot)\|_{L^p_{\text{loc}}}$ is measurable
Below we use Bochner measurability and Bochner integral. Let $T>0$ and $p \in [1, \infty)$. Let $X :=[0, T]$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $...
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Book suggestion on "Banach space geometry for machine learning"
Is there any book for a Mathematics student who can learn Machine learning in the aspect of Banach space geometry? Or, one can understand the connection between Geometry of Banach spaces and Machine ...
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Relation between the distance between two complex measures and their total variations.
Consider measure space $(X,\mathcal{A})$ and space of complex measures on $X$, denoted by
$\mathcal{M}_{C}(X)$. For $\mu \in \mathcal{M}_C(X)$, we define total variation of $\mu$, as $\vert \mu \vert(...
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If $f : \Omega \to X$ is holomorphic, injective, and $f'$ has no zeros, does there exist $\Lambda \in X^*$ s.t. $\Lambda \circ f$ is injective?
This is a follow-up to my latest question: If $f : \Omega \subset \mathbb{C} \to X$ is an injective Banach-valued holomorphic function, can $f'$ ever have a zero?. It was answered concisely and ...
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maximizer of operator norm [closed]
Let $L$ be a continuous linear functional on a Banach space $X$, put $Y=\ker L$. Suppose there is a nonzero $x_0$ in $X$ such that $\|x_0+y\|\ge\|x_0\|$ for all $y\in Y$. Prove that $\|L\|=|L(x)|$ for ...
user1034013
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In the dual to an infinite dimensional Banach space, are the dual vectors defined by how they act on the basis?
Let $V$ be an infinite dimensional Hilbert space and $W$ its dual. Can I identify a $w \in W$ by how it acts on a basis, $B$, of $V$? Does it matter if the dimension of $V$ is uncountable and/or if it'...
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If $f : \Omega \subset \mathbb{C} \to X$ is an injective Banach-valued holomorphic function, can $f'$ ever have a zero?
Let $X$ be a complex Banach space, and let $f : \Omega \subset \mathbb{C} \to X$ be a holomorphic function on an open set $\Omega$. I'll denote by $X^\ast$ the (topological) dual of $X$.
When $X = \...
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Chain rule and duality pairing in Banach spaces
Given a functional $E:Z\rightarrow \mathbb{R}$ defined as
$$
E(z) = G(F(z)),
$$
where
$F: Z \rightarrow H$ is an operator between Banach spaces and
$G: H \rightarrow \mathbb{R}$ is a functional,
the ...
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[Proof verification ]The space $\ell^{2}$ is a Banach space
I would like to know if my proof is correct about the fact that the space $\ell^{2}$ is a Banach space with the usual inner product !
Here is my attempt :
We consider a Cauchy sequence $\{x^{k}\}_{k\...
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A counterexample for the closeness of kernel for linear operator.
Let $ X,Y $ are two Banach space and $ T:X\to Y $ is a linear operator. Assume that the kernel of $ T $, denoted by $ N(T)=\{x\in X:Tx=0\} $ is closed. I want to find a counterexample such that for ...
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A bit confused of a part of a proof of a lemma for the open map theorem (Analysis Now)
Note that $T$ is a linear map that is bounded.
The only part I am a little confused about is the construction of $\{y_n\}$. I understand everything after that. Note, $0 < \epsilon < 1$. I know ...
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If $t_nx_n\to y$ and $A\subseteq [0, \infty)$ is syndetic, is there $r_n\to\infty$ that $r_n\in A$ and $r_nx_n\to y$?
For $A, B\subseteq \mathbb{R}^+$, we denote
$$A+B= \{a+b: a\in A, b\in B\},$$ and
$$A^{-1} +B= \{t: \text{ there is } a\in A \text{ such that } a+t\in B\}.$$
A subset $B\subseteq \mathbb{R}^+$ is ...
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What topology is given to $L(X,Y)$?
Let $E,F$ be two finite-dimensional Banach spaces, with $U\subseteq E$ open and $L(E,F)$ the collection of linear maps $E\to F$. In the following excerpt (from Abraham's Foundations of Mechanics), the ...
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Closure of balanced convex set and its dilation
While reading a proof, one of the steps was showing that for a (specific) balanced and convex set in a Banach space we had the inclusion $\overline{B} \subseteq 2B$. The proof continued using specific ...
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Banach space which has Radon-Nikodym Property +Weakly sequentially complete but not reflexive.
I have been trying to understand reflexivity of a Banach space from geometric point of view. I know that a reflexive space has Radon-Nikodym Property (RNP) and is Weakly sequentially complete(WSC) by ...
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Mathematical Method of Geometric Second Quantization?
I have recently been studying the method of geometric quantization, and I noticed a few methods in it that seem like they could be used to create a geometric second quantization (specifically of the ...
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Is there non-zero vector with bounded orbit if the set of vectors with unbounded is residual and open in Banach space $X$?
Let $T:X\to X$ be a bounded operator on Banach space $X$ such that $A= \{x: \{T^n(x)\}_{n=0}^\infty \text{ is unbounded } \}$ be residual in $X$.
In my research, the set of vector $x\in X$ with ...
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