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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Unitary elements in Banach spaces and subspaces.

Let $F$ be a Banach space and $E$ be a subspace of $F$. Let $e_{0}\in E$ be an element of norm $ 1$ and suppose that span $\{f\in F^{*}:\|f\|=f(e_{0})=1\}=F^{*}$, where $F^{*}$ is the dual space of ...
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Is $\ell_1$ complemented in its double dual $\ell_1^{**}$? (i.e., in $\ell_\infty^*$?)

Quick question, y'all. Is $\ell_1$ complemented in $\ell_1^{**}=\ell_\infty^*$? Yes, I searched Google, and also the standard texts. I can't seem to find an answer, but surely this is known. ...
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26 views

How is Baire category theorem used here?

The following is a doubt that arouse from reading this paper by Bandyopadhyay, Jarosz and Rao. Let $E$ be a Banach space and $E^{*}$ be its dual space. Let $e_{0}$ be an element of norm one in $E$ ...
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26 views

Infinite Hamel basis for Banach spaces

What are some standard examples of Hamel basis for Banach spaces with cardinality >= $\aleph_0$? I tried searching, but couldn't find any.
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Convexity of space of integrals of homotopic paths

Let $B$ be a Banach space. Consider a set $Q \subset B$, a path $\gamma_0:[0,1] \to Q$ and the set of paths $\Gamma_0$ of paths $[0,1] \to Q$ homotopic (with ends fixed) to the path $\gamma_0$. Let ...
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Question about definition of bounded linear functionals

I'm reading through an analysis textbook, and just working through a section on linear functionals. I have the definition: $f$ is a bounded linear functional if $$||f|| = \sup\{|f(x)| : x\in X, ||x|| ...
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25 views

The existence of adjoint operation on Banach space

I have a question about adjoint operator. I have known that bounded linear operator on Hilbert space has a unique adjoint operator, but I am wondering whether there is similar existence result about ...
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30 views

Sufficient condition for a infinite countable or non-countable intersection of open sets is equal to an open set.

Let $(X,\tau)$ a no discrete topological space. If necessary for an affirmative answer consider a metric space $(X, d )$ or a Banach space $(X, \|\,\cdot\, \|)$. In these cases, the topology $\tau $ ...
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Is this space a Banach space? 2

Consider the set of functions $$\mathcal{B}=\{v\in L^2(0,T;H^1_0(\Omega)): \partial_tv\in L^2(0,T;H^{-1}(\Omega))\},$$ equipped with the norm ...
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1answer
38 views

Unconditional basis in $c_0$

We know that in $c_0$ the standard unit vector basis $(e_i)_{i=1}^{\infty}$ is an unconditional basis. For $n\in\mathbb{N}$, let $s_n=\sum\limits_{i=1}^{n}e_i$, my question is that How to prove ...
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Subsequences and blocks of Schauder bases

Suppose $X$ is a Banach space and $(e_n)$ and $(f_n)$ are both Schauder bases of $X$. Does there exist a proper closed subspace $Y\subset X$, and appropriate subsequences of $(x_n)$ and $(y_n)$ that ...
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Integral Measures: Identification

Problem Given a Borel space $\Omega$. Consider a Borel measure: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}:\quad\mu\geq0$$ Regard a Borel measure: ...
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65 views

Equivalent Norms for Intermediate Subspaces

Let $(X,\left\|\cdot\right\|)$ be a Banach space, and let $\left\{T(t) : t\geq 0\right\}$ be an equibounded strongly continuous semi-group on $X$. Define a functional ...
2
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1answer
54 views

Proof of the Banach–Alaoglu theorem

The Banach–Alaoglu theorem states that the closed unit ball of $B'$ (where $B'$ is the dual to a Banach space $B$ over a field) is compact in the weak* topology. I'm having trouble trying to prove the ...
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Showing $X^*$ is separable implies $X$ is separable using the Riesz lemma

If $X$ is a Banach space and $X^*$ is separable, then $X$ is separable. Here, David Mitra mentions a proof using the Riesz lemma. However, I do not fully understand it. You could also use ...
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43 views

Counterexample Of Banach Fixed Point (Banach's Contraction) Theorem

Banach Fixed Point theorem states: Let $(X,d)$ be a complete metric space. Suppose that $f:X→X$ is a strong contraction, i.e. there exists $q ∈ [0, 1)$ such that $d(f(x),f(y))$ $\le$ $q$ $d(x,y)$, ...
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To be or not to be Banach? That is the question.

On the set $H^1_0((0,2))$ we put the following norms. $$\|u\|_a^2= \int_{[0,2]}(u')^2.$$ $$\|u\|_b= \|u\|_\infty.$$ $$\|u\|_c= \|u\|_{L^2}.$$ Is $H^1_0((0,2))$ Banach with any of these norms?
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Extending finite rank operators

Suppose $Y$ is a closed subspace of Banach space $X$ and $T:Y\to X$ is a bounded finite rank operator. Can we extend $T$ to $\tilde{T}:X\to X$, in the sense that: $T=\tilde{T}$ on $Y$ ...
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1answer
34 views

Trouble understanding Banach limit in Stone-Cech compactification of $\mathbb{N}$

Trouble understanding Banach limit in Stone-Cech compactification of $\mathbb{N}$. For example, if I have a series $\{a_n\}_{n\in \mathbb{N}} \in [0,1]$, what does it mean that the limit of the series ...
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Can We Always Realize the Value of the Quotient Norm. [duplicate]

Let $(V, \|\cdot\|)$ be a Banach space over $\mathbf R$ and $W$ be a closed subspace of $V$. We know that $V/W$ becomes a normed linear space under the quotient norm $\|\cdot\|_q$ defined as ...
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1answer
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Show $l^p$ embeds in $L^p(0,1)$

Let $l^p$ be the standard sequence space indexed by $\mathbb N$. I've heard it claimed that $l^p$ embeds into $L^p(0,1)$ in such a way that $$L^p(0,1)=l^p\oplus S$$ for some closed subspace $S\subset ...
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1answer
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$C^1(\bar \Omega)$ is a Banach space

My professor gave a proof of the completeness of $(C^1(\bar \Omega),\|\cdot \|_{C^1})$ based on the fundamental theorem of calculus. I though about an alternative and I would like to know whether this ...
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Is every vector space a banach space?

Using the axiom of choice one can show that for each ($\mathbb{R}$-) vector space $V$ there exists a function $\|\cdot\| : V \rightarrow \mathbb{R}$ so that $(V,\|\cdot\|)$ is a normed vector space. ...
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Hahn Banach Theorem: Clarification on meaning of extending a functional?

Hahn Banach Theorem: Given linear (vector) space $\mathbb{X}$, define $u \in \mathbb{L} \subset \mathbb{X}$, $A,B,C$ functionals, A sublinear. $A:\mathbb{L} \to \mathbb{R}, B:\mathbb{L} \to ...
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1answer
55 views

Clarification on the difference between Brouwer Fixed Point Theorem and Schauder Fixed point theorem

From Zeidler's Applied Functional Analysis Brouwer The continuous operator $A:M \to M$ has a fixed point provided $M$ is compact, convex, nonempty set in a finite dimensional normed space ...
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If $\dim X=n$ then for any norm in $X$, $X$ is complete. [duplicate]

I know there are standard proofs for this theorem, but I need to prove it by contradiction or proving that $\dim X=\infty$. I thought maybe using Hahn-Banach? Thanks.
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1answer
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Weakest condition on $f\colon \Bbb R^2\to \Bbb R$ so that $f(\|x\|_1,\|x\|_2)$ is a norm.

$\newcommand{\norm}[1]{\|#1\|_1}\newcommand{\morm}[1]{\|#1\|_2}\newcommand{\xorm}[1]{\|#1\|_3}$ Let $X$ be a finite dimensional Banach space and $f\colon \Bbb R^2\to \Bbb R$. What is the weakest ...
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Contraction: Precompactness

Given Banach spaces $X$ and $Y$. For precompactness: $$\tau\in\mathcal{C}(X,Y):\quad\overline{A}\text{ compact}\implies\overline{\tau(A)}\text{ compact}$$ Is this true and why?
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Dual of $C(K,X)$, for scattered $K$

Let $K$ be a compact, Hausdorff space and $X$ be a Banach space. By $C(K,X)$ we denote the Banach space of all continuous functions $f : K \to X$, equipped with the supremum norm: \begin{align} ...
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What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| ...
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$A^2$ has a fixed point implies $A$ has also a fixed point

Let $X$ be a Banach space and $A:X\to X$ a bounded operator. Assume that $A^2$ admits a fixed point in $X$ i.e. there exists $x_0\in X$ such that $A^2x_0=x_0$. Does this mean that $A$ has also a fixed ...
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1answer
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Finite Rank Operator: Continuity

I keep forgetting it, so... Given Banach spaces $X$ and $Y$. Then it is wrong: $$\dim\mathcal{R}F<\infty:\quad F\in\mathcal{L}(X,Y)\implies F\in\mathcal{C}(X,Y)$$ Can I construct such?
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Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
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Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying ...
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3answers
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Convergence on the dual of a Banach space

I have a simple question : What it means $$||v_n||_{(W^{1,p}_0)^*}\rightarrow 0$$ Where $(W^{1,p}_0)^*$ is the dual space of $W^{1,p}_0$ I know that $v_n\rightarrow 0$ in $(W^{1,p}_0)^*$ mease ...
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1answer
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Dense Domain: Preimage

Given Banach spaces $X$ and $Y$. Regard a bounded operator: $$A\in\mathcal{B}(X,Y)\implies A\in\mathcal{C}(X,Y)$$ Then for dense sets: $$W\leq Y:\quad \overline{W}=Y\implies\overline{A^{-1}W}=X$$ ...
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Closed subsets of Banach space

$X=\{(a,b)\mid a \in C[0,1],b \in C[0,1]\}$, and its norm is $\|(a,b)\|=\|a\|_\infty+\|b\|_\infty.$ $Y=\{(a,a')\mid a \in C^1[0,1], \ a'(t)=\frac{da}{dt} \},\ Z=\{(0,b)\mid b \in C[0,1]\}.$ ...
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1answer
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Is $C[0,1]$ equipped with $\lVert \cdot \rVert_1$ a countable union of nowhere dense sets?

Let's consider the space of all continuous function $C[0,1]$ on the intervall $[0,1]$. But instead of using the usual supremum norm we use the $L^1$-Norm $: \lVert f \rVert_1=\int_0^1 \lvert f(x) ...
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Dense Operators: Spectrum

This thread is Q&A. Given a Banach space $E$. Consider closed operators: $$T:\mathcal{D}(T)\subseteq E\to E:\quad T=\overline{T}$$ Then for the domain: ...
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Does for $T \in B(X)$ with $\|T\|>1$ exist $T^{-1}$?

Is it true if $\|T\|>1$, where $T \in B(X)$ for some Banach space $X$, then $T^{-1}$ exists? I suppose that for $\|T\|=1$ this isn't true? Because, if we suppose that inverse exists for such ...
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Please give me an example of closed subspace of banach space under some conditions

Please give me one example of Banach space $X$ and its closed subspaces $S,T,U$ which suffice following conditions. Any of $S+T,T+U,U+S$ is not a closed subspace of $X$. I can say there are some ...
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Convergence of sequence as $O(1/n)$ using damped iterations

I am trying to understand the argument for convergence here (page 16) for damped iterations of non-expansive maps. Say we have a sequence $\{x^n\}_{n=0}^{\infty}$ that is generated as $x^n = \theta ...
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When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
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Fixed-point analysis similar to Banach Fixed Point Thm

I have a fixed-point question similar to the Banach fixed-point theorem. Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a given function and let $x^\star \in \mathbb{R}^n$ be a known ...
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Uniform Boundedness: Nets

I thought so far that the uniform boundedness principle applies (according to the proof I know) to any net of bounded operators. Now I read that this works for sequences only. Can you shortly explain ...
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Summability: Equivalence

Summability Given a Banach space $E$. Consider sums: ...
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Do we need completeness for a weak*-convergent sequence to be bounded?

Let $(\phi_n)_n$ be a weak* convergent sequence in the dual of some normed space $X$ with (weak*-)limit $\phi$. If $X$ is Banach then it follows from the uniform boundedness principle that $\sup_n ...
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Norm of a product of projections in a Banach space

Let $X$ be a Banach space and let $P_1,P_2$ be two projections in $B(X)$, i.e., $P_1^2 = P_1, P_2^2=P_2$. My question: under what conditions do we have that $\Vert P_1 P_2 \Vert = \sqrt{\Vert P_2 ...
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Equivalent definition of uniform convexity

A Banach space $X$ is said to be uniformly convex if the following is satisfied: For $\epsilon>0, \exists \delta>0$ such that $x,y\in X, \|x\|, \|y\|\leq 1$, $\|x-y\|\geq \epsilon \Rightarrow ...
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What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't ...