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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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Proof of ABCD Lemma

A standard result in bifurcation theory due to Keller, known as ABCD Lemma, asserts that if $V$ is a Banach space, $M\,:\,V\times \mathbb{R} \to V\times \mathbb{R}$ is such that $$ M:=\begin{pmatrix} ...
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1answer
28 views

Dependence and Independence of set of functionals [on hold]

Let $X$ be a Banach space on over $\mathbb{R}$ and let $\psi$ and $\phi$ be two linear functionals defined on $X$. Then, if $\ker(\phi)\subset \ker(\psi)$, can we have that the set $\{\phi, \psi\}$ is ...
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25 views

Only zero in kernel of every element of a subspace of the dual does not imply density of said subspace for non-reflexive Banach spaces

Let X be a Banach space and $M \subset X'$ a subspace of its dual space. If X is reflexive we know the following statements are equivalent: (i) $M$ is dense in $X'$ (ii) $x \in X$ and $\phi(x)=0$ ...
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48 views

Converse of Schauder's Theorem about compactness of adjoint operator

It is well known (also known as Schauder's Theorem) that if $X$ and $Y$ are normed spaces and $T:X\to Y$ is a linear and compact operator, then also $T^*:Y^*\to X^*$ is compact. The converse is true ...
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Finite-dimensional copies of an unconditional basis uniformly complemented in $(\oplus\ell_\infty^n)_p$

Fix $1\leqslant p<\infty$, and let $(e_n)_{n=1}^\infty$ be an unconditional Schauder basis with some special properties which I'll discuss below. I'm trying to prove the following conjecture: ...
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1answer
64 views

$||\phi_z - \phi_w||_{A(\mathbb{D})^*}= 2$ if and only if $z$ or $w$ in $\mathbb{T}$

Let $\mathbb{D}$ be the open unit disc in the complex plane $\mathbb{C}$. Let $A(\mathbb{D})$ stand for the space of all continuous functions on $\overline{\mathbb{D}}$ which are holomorphic in $\...
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29 views

Applying Geometric Hahn-Banach Theorem on Vector-Valued Integration

I have been studying the applications of geometric Hahn-Banach Theorems. I came across this book Functional Analysis by S. Kesavan where the geometric Hahn-Banach Theorem is applied on Vector-Valued ...
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Frechet Space vs Banach Space

What is Frechet Space? Is a Banach a Frechet space? If not, why? If yes, how do we prove it? According to Wikipedia, Frechet spaces are locally convex topological space that is complete with ...
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1answer
27 views

Spectral theorem for compact operators on Banach space.

Let $X$ be a Banach space. Let $A$ be a compact operator on $X$ and let's denote $\sigma(A)$ is spectrum of operator $A$. Let $f$ be holomorphic function in some neighbourhood of $\sigma(A)$ Out ...
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38 views

Show that $(X, \vert\vert\vert \cdot \vert \vert \vert)$ is a Banach Space and that $(X, \vert \vert \cdot\vert \vert_{1})$ is not)

Let $X:=\{ x \in \ell^{1}:\vert\vert\vert x \vert \vert \vert< \infty\}$, and that $\vert\vert\vert x \vert \vert \vert=\sum\limits_{j=1}^{\infty}j\vert x_{j}\vert$ $a.$ Show that $(X, \vert\vert\...
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Banach space exercise

Suppose $X$ and $Y$ are Banach spaces and $T_{n} \in B(X,Y)$. If $T_{n}x_{n} \to 0$ in $Y$ for any choice of unit vectors $\{x_{n}\}$ in $X$, show that $\|T_{n}\| \to 0$.
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$s$-convexity implies $p$-convexity in Banach lattices

If $1<p<s<\infty$ and $E$ is a Banach lattice which is $s$-convex, is it also $p$-convex? If so, what would be a good reference for this?
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47 views

Right Hilbert space for a duality pairing

Let $B$ be a Banach space. Let $B^*$ be its dual, and let $K\subseteq B^*$ be some linear dense subspace. Denote the duality pairing between $B$ and $B^*$ via $\langle\cdot ,\cdot \rangle$. Suppose we ...
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37 views

Compactness when mapping into a higher $L^p$ space and then back

Question: Let $q>p \ge 1$ and let $T:L^p[0,1] \to L^q[0,1]$ be a bounded linear operator. Let $i: L^q[0,1] \to L^p[0,1]$ be the inclusion map (which is bounded). Is the composition $i\circ T$ ...
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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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47 views

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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1answer
78 views

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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30 views

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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31 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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47 views

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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1answer
29 views

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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1answer
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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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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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34 views

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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37 views

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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35 views

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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16 views

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
22 views

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
31 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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21 views

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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2answers
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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1answer
34 views

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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1answer
41 views

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
25 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^{**} $ ...