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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homeomorphism of a Banach space, constructed using a contraction

Let $X$ be a Banach space and $g: X \to X$ a contraction, meaning that for all $x, y \in X$, we have that $$||g(x) - g(y)|| ≤ L ||x - y||$$ for a constant $0 ≤ L < 1$. Now, consider the function ...
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Show that $C^{1}([a,b];\mathbb{R})$ is a Banach space.

Let $C^{1}([a,b];\mathbb{R})$ the vectorial space of the functions (bounded) $f:[a,b]\to\mathbb{R}$ where all $f$ has a continuous derivate (and bounded) in all point of $[a,b]$, with the norm ...
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If a functional is bounded below must it have a minimizing sequence? [on hold]

Let $X$ be a Banach space. If $\mathcal{I}:X\rightarrow \mathbb{R}$ and bounded below, must there exists a minimizing sequence, i.e., $\{u_k\}_{k=1}^{\infty}\subset X$ such that ...
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How to show that every Cauchy sequence converges in some normed linear space?

(Hunter and Nachtergaele, 1.6) Using the fact that R is a complete metric space with respect to Euclidean distance, show that Rn is a Banach space when equipped with (a) the Euclidean norm (b) the ...
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The weak topology on an infinite dimensional linear space is not first-countable

I thought I needed help proving the above statement, but during typing I found a proof. Since I had already written it all down I will post it anyway, maybe in the future someone can benefit from it. ...
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Embedding the Schreier spaces into $C(\alpha)$ for some countable ordinal $\alpha$

Let $1\leq\xi<\omega_1$ be any countable ordinal, and denote by $\mathcal{S}_\xi$ the Schreier family of order $\xi$. Then the Schreier space of order $\xi$ is the completion $S_\xi$ of $c_{00}$ ...
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The gradient of a function on a Banach space is an element of the dual space

Can somebody explain me why gradient descent in Banach space does not make sense? As pointed out by Sebastien Bubek in his blog, the gradient is an element of the dual space $\mathcal{B}^*$. But I ...
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Prove $2a^Tb \leq \|a \|^2 + \|b\|_*^2$ with dual norm

We know we can easily (**) prove $$ 2a^Tb \leq \|a \|_2^2 + \|b\|_2^2, \forall a,b $$ Is there a way to prove the following: $$ 2a^Tb \leq \|a \|^2 + \|b\|_*^2, \forall a,b $$ where ...
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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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28 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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27 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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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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40 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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45 views

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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49 views

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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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 ...
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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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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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55 views

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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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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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 $S\subset ...
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$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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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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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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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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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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The tensor product of $ {L^{1}}(G) $ and a Banach space

Let $ G $ be a locally compact group and $ A $ a Banach space. It is known that the tensor product $ {L^{1}}(G) \otimes A $ is isometrically isomorphic to $ {L^{1}}(G,A) $. I need a proof of it.
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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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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 ...