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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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Linear operator in banach spaces: multiplication of open balls by positive scalar

I would like to demonstrate the following. Let $X,Y$ be Banach spaces and $T:X \rightarrow Y$ be a linear operator. Hyposthesis: Suppose you can take an open ball in Y such that $B(y, \epsilon) \...
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Grothendieck's lemma in $L^p$ spaces

So I am currently working on the proof of Grothendieck's Lemma : Let S $ \subset L^{\infty}(X) $, of finite measure, be a closed vector subspace of $L^p $ for a certain p such that $ S \subset L^{\...
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Proving that the set of continuous linear maps (From a Banach space $X$ to another Banach space $Y$) is an open subset of $L(X,Y)$

I am a university undergraduate student, and I am reading up on the German Functional Analysis textbook "Introduction to Functional Analysis" by Friedrich Hirzebruch/Winfried Scharlau. My question ...
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Definition - A Banach space continuously embedded in the space of distributions on the unit circle

What is meant by a Banach space continuously embedded in D'( $\mathbb{T}$ ), where D'($\mathbb{T}$) denotes the space of distributions on the unit circle, and how is such a space constructed? Why ...
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Convergence of a series with general index set

I saw that it is possible to define the convergence of a sequence indicized by a general set (with an order relation) using NETS. My question is if it is possible to define in this way the notion of ...
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$(C^1[0,1], || . ||_\infty)$ is not Banach Space.

Recently, I am studying Banach Space. I know that $(C[0,1], || . ||_\infty)$ is a Banach Space and closed subspace of Banach Space is Banach. This follows from the result that closed subset of ...
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28 views

A question regarding Radon-Riesz property

I want to show that if a normed linear space has Radon-Riesz property, then it has the semi Radon-Riesz property. We know that a normed linear space is said to have Radon-Riesz property if for every ...
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Quotient space of a a normed linear space with schur's property need have have that property.

I want to show that if a normed linear space has Schur's property (every weakly convergent sequence converges), then it is not guaranteed that every quotient space $X/Y$ will have that property, where ...
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Is there a norm making $C([0,1])$ into a Hilbert space?

The space $C([0,1])$ of continuous functions on $[0,1]$ is an inner product space under the $L^2$-norm, but not complete. Equipped instead with the $L^\infty$-norm, it becomes complete but the norm is ...
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Books about fixed-point theory

I am looking for book recommendations about Fixed-Point Theory. I found this post: Book Recommendation for Iterated Functions? recommending a book by Shashkin. However, I cannot find who Shashkin is,...
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If $T:L^p[0,1] \to L^p[0,1]$ bounded for $1 < p < \infty$ with continuous image, then it's compact

Is the following statement true? Let $T:L^p[0,1] \to L^p[0,1]$ be a bounded operator for $1 < p < \infty$ and suppose that $\operatorname{Im}(T) \subset C[0,1]$ consists of continuous functions....
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Complement a finite dimensional subspace in a Banach space

Given a Banach space $(X,\|,\|)$, and a finite dimensional subspace $F \subset X$, is it always possible to choose a closed linear complement to $F$. Explicitly, I mean to say, will there always exist ...
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From $\sup_{n\in \mathbb{N}} \left|\int_0^\Lambda e^{nx} f(x) dx \right| < \infty$ to $f\equiv 0$

Given $f\in C[0,\Lambda]$ satisfying $$\sup_{n\in \mathbb{N}} \left|\int_0^\Lambda e^{nx} f(x) dx \right| < \infty$$ Prove that $f\equiv 0$ $\,\forall x\in[0,\Lambda]$ I found a weaker ...
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Counterexample for “continuous image of closed and bounded is closed and bounded” (in normed spaces).

It's well known that: If $X$ is a finite-dimensional normed space, $C$ is a closed and bounded subset of $X$ and $f:C\subset X\to X$ is continuous, then $f(C)$ is closed and bounded. If $X$ is any ...
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Alternative proof of Taylor's formula by only using the linear approximation property

So a function $f: E \to F$ between the normed spaces $E,F$ is called differentiable in $x \in E$ if there exists a bounded linear map $Df(x): E \to F$ such that for every $h \in E$ we have $$f(x+h)=f(...
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Sequences of Continuous Linear Operators between Banach Spaces.

Let $E, F$ be two Banach Spaces. Let $\{ T_{n} \}$ be a sequence of continuous linear operators from $E$ into $F$ such that: For all $x \in E: T_{n}x \rightarrow Tx,$ some limit in $F$. Then the ...
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Find the Spectra of this Bounded Linear Operator

Let $X$ denote the space $L^{1}(\mathbb{R})$ of all (equivalence classes of) Lebesgue integrable functions $f:\mathbb{R} \to \mathbb{C}$ with the norm $||f||_{1} = \int_{\mathbb{R}}|f(t)|dt$. Let $T \...
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64 views

Convex, absorbing sets and nonempty interior

Let $A$ be a convex, absorbing subset of a real Banach space $X$ with the additional property that the closure $\rm{cl}(A)$ contains an open ball around $0\in X$. Does this imply that already $A$ ...
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Showing that $A(\overline{B_1^X}) \subseteq Y$ is closed when $A \in \mathcal{L}(X,Y)$.

Exercise : Let $X,Y$ be Banach spaces and $X$ be reflexive, $A \in \mathcal{L}(X,Y)$. If $\overline{B_1^X} = \{ u \in X : \|u\|_X \leq 1\}$, show that $A(\overline{B_1^X}) \subseteq Y$ is closed. ...
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Characterization of Reflexive Banach Space.

Prove that a real Banach Space $X$ is reflexive if and only if each pair of disjoint closed, convex subsets of $X$, one of which is bounded, can be strictly separated by a hyperplane. The theorem is ...
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Under what conditions does $L^{1}(X)$ have a predual?

I know this question has been asked a million times—but they seem to always be with some special flair. I've looked at many and cannot extract from them an answer to my plain question: Question: Let $...
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functional analysis : problem related to closed graph theorem

enter image description here the problem above is in Conway's [Functional Analysis] (p.93) it seems to be an application of closed graph theorem if the inequality were posed the other way it could ...
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Uncountable basis in Hilbert space vs orthonormal basis

It is known that an infinite dimensional Banach space does not have a countable Hamel basis. It is also known that a separable Hilbert space has an orthonormal countable basis. Now, I think this basis ...
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implicit function theorem, necessary condition for bifurcation point

I want to derive a necessary condition for $\lambda^*$ to be a bifurcation point. Some context to the problem I am studying: Let $F \in C^2(\mathbb R \times X) \; \; ,F:\mathbb R \times X \...
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1answer
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If Banach space $C(K)$ is decomposable (non trivial) then $C(K)$ is isomorphic to some of the sumands subspaces

In Beauzamy (Banach spaces) book appears this statement without proof: "if $X\oplus Y$ is isomorphic to Banach space $C(K)$ then either $X$ or $Y$ is isomorphic to $C(K)$'' where $X$, $Y$ are ...
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1answer
62 views

Duality Theorem Confusion

A book that I am reading states the following theorem: Theorem. Let $x$ be an element in a normed linear space $X$ and let $d$ denote its distance from the subspace $M (\bar{M}\neq X)$. Then \begin{...
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1answer
28 views

prove an operator is compact in reflexive space

In Banach spaces $E,F$, a compact operator $T\in \mathcal{L}(E,F)$ maps weakly convergent sequences into strongly convergent sequences. If E is reflexive, the converse is true. I need help in proving ...
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Strict inequality for the norm of a multiplication operator

Let $S\neq\emptyset$. We call a Banach space $X\subset\mathbb{C}^S$ a Banach functional space if evaluation at each point is a bounded linear functional, i.e. $e_s:X\to\mathbb{C}$ with $e_s(f)=f(s)$ ...
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Banach Isomorphism Theorem application

Let X be a Banach Space and $L, M$ closed subspaces such that $L\cap M = \{0\}$. I would like to apply Banach Isomorphism Theorem to prove that if $L+M$ is closed, then $P : L + M \to L$ defined as $P(...
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1answer
49 views

$A \in \mathcal{L}(X,Y) \implies A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$

Exercise : Let $X,Y$ be Banach spaces and $A \in \mathcal{L}(X,Y)$. Show that $ A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$. Attempt : The linearity is trivial. Τo show that $A^*$ is $w^*$ to $w^*$...
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Approximating a Banach space valued function by sums of continuous functions

I am trying to prove the following exercise, which is a part of a project type homework problem. Please give hints and suggestions, and discuss this problem. Let $(T,d)$ be a compact metric space, ...
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26 views

convex subset of topological dual

Let $(E, ||.||_E )$ be a banach space, and $E^∗$ its topological dual. For $u ∈ E$, prove that $F(u) =\{L\in E^*, ||L||_{E^∗} = ||u||_E, \left<L, u\right> = ||u||^2_E \}$ is convex. Let $t\in ...
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Exercise about a Corollary of the Uniform boundedness principle

Let $T_{jk}$ be bounded operators from $X$ to $Y$, where $X$ and $Y$ are Banach spaces. Prove that if $\forall j$ $$\sup_{k} ||T_{jk}||=\infty$$ then there exists a vector $x \in X$ s.t. $$\forall j, \...
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Does $L^p(E)$ with $m(E)<\infty$ with smaller norm preserve the Banach

If $1\leq p < q <\infty$ and $E$ a subset of $\mathbb{R}$ with finite measure if we consider the space $L^q(E)$ is it a Banach space with the norm $||.||_p$. I know that $L^p$ space is a Banach ...
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1answer
24 views

Example of a topological space $X$ such that $C_0 (X)$ is not a $C^*$-sub-algebra of $C^b (X)$

Let $X$ be an arbitrary topological space. If $X$ is locally compact and Hausdorff, then $C_0 (X)$ (space of continuous functions vanishing at infinity) is a $C^*$-sub-algebra of $C^b (X)$ (space of ...
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Boundedness of a Net in Second Dual Transfering to Original Space

Let $ X $ be a Banach space. Let $ J : X \to X^{**} $ denote the natural embedding of $ X $ into $ X^{**} $. Suppose that there exists a bounded net $ (T_{\alpha})_{\alpha \in I} \subset X^{**} $ ...
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1answer
24 views

Closed ideal in $ L^{1}(G)$

Let $G$ be locally compact group prove that $$L_{0}^{1}(G)=\left\{f\in L^{1}(G): \int_G f(g) dm(g)=0 \right\}$$ is a closed ideal in $ L^{1}(G)$ with codimension one I am grateful for any ...
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1answer
44 views

Why are these two characterizations of compact operators equivalent?

If $T$ is a bounded linear operator on a Hilbert space $H$, then I have heard that the following two things are true: $T$ is compact if and only if $T(C_1)$ is compact where $C_1$ is the closed unit ...
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Prove $ C^1[0,1]$ is not a Banach space with the sup-norm [duplicate]

Let $ C^1[0,1]$ be the normed space of continuously differentable functions on $[0,1],$ with $||x||=\max_{t\in [0,1]} |x(t)| $. Prove that $ (C^1[0,1],||.||)$ isn't a Banach space I think we need ...
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63 views

Is this space equivalent to the James space?

The James space $J$ is a famous counter-example in functional analysis. It is an example of a Banach space that is isometrically isomorphic to its double dual, but is not reflexive. Define $$J = \...
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$\ell^p(J)$ is complete (Banach)

I have to prove that the space $\ell^p(J)$ defined as the set of all functions $\psi: J\rightarrow \mathbb{F}$ s.t. $\psi$ is null except in a contable subset of $J$ and $||\psi||_p :=\bigg(\...
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1answer
55 views

If $T$ is a topological isomorphism, then so is $T^*$

The question comes from the following link on page 25: https://www.ucl.ac.uk/~ucahad0/3103_handout_3.pdf They prove $(T^{-1})^*=(T^*)^{-1}$, but I don't see how it proves $T^*$ is a topological ...
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Let $X$ be a Banach space. $P∈B(X)$ satisfies $P^2=P$. Prove that $X=\ker P⊕P(X)$ [closed]

Let $X$ be a Banach space. $P∈B(X)$ satisfies $P^2=P$. How to prove that $X=\ker P⊕P(X)$ ? Thanks for your help!
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Approximating a Banach space “vector- valued function” by “simple functions”

I'm trying to prove the following claim: Let $T$ be a compact (metric) space, and let $\mathcal{X}$ be a Banach space over $\mathbb{K}$. Let $f : T \longrightarrow \mathcal{X}$ be a continuous ...
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Norm $C^1$ multivariate functions

Given an open and bounded subset $\,\Omega\subset\mathbb{R}^n\,$ and the interval $\,[0,T]\,$ such that $\,T<\infty$, I'm trying to find a norm for $\,X=\mathcal{C}^1\left([0,T]\times\Omega\right)$ ...
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1answer
22 views

The relationship among different types of fundamental spaces.

I'm just looking to make sure my understanding of certain fundamental spaces are correct. Denote the set of all vector spaces by $V$, the set of all metric spaces by $M$, the set of all normed spaces ...
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Every Dunford-Pettis operator is compact

I was trying this problem. Let $X$ be a Banach space that does not contain a copy of $l^1$. Show that every Dunford-Pettis operator $T: X \rightarrow Y$, with $Y$ any Banach space, is compact. I ...
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Is $X$ reflexive if $X=\cup Y_i $ and $Y_i$ is reflexive for every $i$?

Let $X$ be a Banach space. Assume that there are a family of closed (with respect to $\left\|\cdot\right\|_X$) subspaces $Y_i$ of $X$ such $X=\cup Y_i $ and $Y_i$ is reflexive for every $i$. Can we ...
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1answer
54 views

Uniqueness of Hahn-Banach extensions for $c_0 \subseteq \ell^\infty$

Let $c_0$ be the space of real sequences converging to zero with supremum norm. $c_0$ is a (closed) subspace of $\ell^\infty$, the space of bounded real sequences. A $f \in {c_0}^*$ corresponds to a $...
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1answer
23 views

Convergence of a sequence by the convergence of a subsequence

Let $X$ be a Banach space and $\{x_{n}\}_{n\in\mathbb{N}}$ be a bounded sequence in $X$. Assume for any subsequence $\{x_{n_{k}}\}_{k\in\mathbb{N}}$ of $\{x_{n}\}_{n\in\mathbb{N}}$, there exists a ...