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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$x^*\circ f:G\rightarrow \Bbb{C}$ is analytic. Show $f$ is analytic.

This is an exercise in Conway's 《A course in Functional Analysis》. X is a complex Banach space. $G$ is an open set in the complex plane and $f:G\rightarrow X$ is a function such that for each $x^{*}$ ...
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19 views

Relation between homomorphisms and Lipschitz functions

Given a Riemannian (say, connected) manifold $(M,g)$ one can produce the metric via a formula $d(x,y)=\inf_{\gamma}l(\gamma)$ where $\gamma$ is piecewise smooth curve joining $x$ and $y$. There is ...
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2answers
68 views

Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$?

Let $X, Y$ be Banach spaces, $S \in B(Y^{*}, X^{*})$. Does such operator $T \in B(X, Y)$ exist so that $T^{*}=S$? I suppose that the answer should be - no. Are there any hints that might help in ...
2
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1answer
40 views

Complex of banach spaces is exact if and only if its dual is exact

Let's consider two complexes of Banach spaces: $ X \rightarrow Y \rightarrow Z$, with the maps $S: X \rightarrow Y$, $T: Y \rightarrow Z$. The dual complex looks like $Z^{*} \rightarrow Y^{* } ...
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23 views

1st isomorphism theorem on linear transformations between Banach spaces

First isomorphism theorem: Let V,W be vector spaces and $f:V\to W$ be modules homomorphism then $V/ker(f)\cong Im(f)$. On the other hand, by an exercise in Folland's "Real Analysis" book (chapter 5, ...
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1answer
67 views

If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$?

$U$ is a bounded open subset of $R^n$. If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$ ?
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1answer
36 views

Showing codimension of subspace of C[0,1] equals 1

Show that $\overline{\{f∈C^1[0,1]:f(0)=0\}}$ as a subspace of $C[0,1]$ has codimension 1. Attempt: define $T:C[0,1]\to$ $\Bbb{R}$ by $T(f)=f(0)$. $T$ is a surjective continuous linear ...
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15 views

Differentiation of a continuous bilinear form

Let $E_1, E_2$ and $F$ be $3$ Banach spaces. Let $B: E = E_1 \times E_2 \to F$ be a continuous bilinear form. Show that $B$ is differentiable at every point $a = (a_1,a_2) \in E$ and its differential ...
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25 views

Does it necessarily follow that $C^{n, \gamma}(\overline{X})$ is a Banach space? [duplicate]

Let $X$ denote an open subset of $\mathbb{R}^n$. Suppose $n \in \{0, 1, \dots\}$, $0 < \gamma \le 1$. Does it necessarily follow that $C^{n, \gamma}(\overline{X})$ is a Banach space?
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Weak Convergence Inequality

Let $X$ be a Banach space and $X^*$ it's dual space. a) If $\left\lbrace x_n\right\rbrace$ converges weakly to $x$ in $X$, then $sup_n \|x_n\| < \infty$ and $\liminf_n \|x_n\| \geq \|x\|$ ...
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2answers
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Showing a set is closed and nowhere dense in a Banach space

Let $X$ and $Y$ be Banach spaces, $T_j \in L(X,Y)$ for each $j$ and let $E_n = \left\lbrace x \in X: \sup_{j \geq 1} \|T_jx\| \leq n\right\rbrace$. Show $E_n$ is closed for each $n$ If ...
2
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1answer
67 views

Closed Graph Theorem Application

I'm having trouble working out the proof for this problem. Let $T:L^2(X)\to L^2(X)$ be a linear map such that there is another linear map $T^*:L^2(X)\to L^2(X)$ with $\langle Tu,v\rangle=\langle ...
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1answer
39 views

How to show that a Banach space is a subspace of another Banach space?

I'm having some trouble showing that the following Banach space is a subspace of $\ell^1(\mathbb{N} )$ $\ell^1_w(\mathbb{N} ) = \{ \{x_k\}^\infty_{k=1}| x_k \in \mathbb{C}, \sum_{k=1}^{\infty} ...
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1answer
51 views

Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a Banach space for any norm $\|\cdot\|$ on it?

Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a complete space for any norm $\|\cdot\|$ on it ?
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Can any uncountable dimensional real vector space be made into a Banach space?

On any real vector space $V$ of uncountable dimension , can we always define a norm such that endowed with that norm , $V$ becomes a complete normed linear space ? ( I know it can be done if $V$ is ...
2
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1answer
23 views

Basic properties of Riesz spaces

I'm self studyhing from Peter Meyer-Nieberg's Banach Lattices, and I'm having some trouble with some of the very basic properties. So, what I have to work with at this point is the definition: We ...
2
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1answer
61 views

$X$ be a real normed linear space ; if $\mathcal L(X,X)$ is complete then is $X$ also complete?

Let $X$ be a real normed linear space and $\mathcal L(X,X)$ denote the set of all bounded linear operators on $X$ , we know that if $X$ is complete then so is $\mathcal L(X,X)$ ; is the converse true ...
5
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1answer
54 views

Are all operators to or from $\ell_1$ completely continuous?

Let $E$ and $F$ be two Banach spaces, and let $T \in \mathcal{L}(E, F)$. Consider the following property (P). For every weakly convergent sequence $(u_n)$ in $E$, $u_n \rightharpoonup u$, then ...
2
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1answer
71 views

Can all compact subsets of $\mathbb{C}$ be spectra of a bounded operator on $C[0,1]$?

Let $K$ be a non-empty compact subset of $\mathbb{C}$, the complex field. Does there exist an operator $A\in \mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is ...
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1answer
55 views

Proving something is a Banach Space

Prove that $(\ell ^∞,||·||_∞)$ is a Banach space using the following steps. Let $(x_n)_{n∈\mathbb N}$ be a Cauchy sequence in $(\ell ^∞,||·||_∞)$. For $n > 1$, let $x_n = ...
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1answer
22 views

The w*-extension of a bounded linear functional

Let Y be a Banach space and assume that $X$ is a $w^*$-closed subspace of $Y^*$. Let $f$ be a bounded linear functional on $X$. Does there exist any $w^*$-continuous linear functional $\phi$ on $Y^*$ ...
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1answer
34 views

If $\|\cdot\|_{1}\le\|\cdot\|_{2}$ then $\|\cdot\|_{2}\le M\|\cdot\|_{1}$ [duplicate]

Let $(X,\|\cdot\|_{1})$ and $(X,\|\cdot\|_{2})$ be complete normed vector spaces and $\|x\|_{1}\le\|x\|_{2}$ $\forall x\in X$. I want to prove that $\exists M>0$ such that $\|x\|_{2}\le ...
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1answer
59 views

The algebras of compact operators on $\ell_p$ as direct limits of matrix algebras

Consider $M_n(\mathbb{C})$ as $B(\ell_p^n)$ for $n\in\mathbb{N}$ where $p\in[1,\infty)$, and include $M_n(\mathbb{C})$ in $M_{n+1}(\mathbb{C})$ as the upper left corner. Is it true that ...
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How do I get $\|x\|\le C\|y\|$ in this case?

I feel that the title is a bit uninformative, please feel free to edit it. This is a problem related to the Open Mapping Theorem. Let $T:X\to Y$ be a bounded linear operator from a Banach space X to ...
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2answers
107 views

Is $C[0,1]$ reflexive?

I.e. is the embedding $C[0,1]\hookrightarrow \left( C[0,1] \right)^{**}$ surjective? I am having a hard time answering that question. It would be enough to find a closed subspace of $C[0,1]$ which ...
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1answer
24 views

Application of uniform boundedness principle for sequence of operators

Let $T_n:X:=C([0,1])\rightarrow C([0,1])=:Y$ be a sequence of linear operators. Suppose: $T_n$ is bounded $(Tf)(x):=\lim\limits_{n\to\infty}(T_nf)(x)$ exist for every $f\in C([0,1])$ and $x\in ...
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25 views

Infinite dimensional $\sigma$-compact space

It is well-known that Banach space is $\sigma$-compact iff it is finite dimensional. I'm looking for examples of infinite dimensional normed $\sigma$-compact spaces.
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Weak convergence in Banach space

If $X$ is a Banach space, $X'$ its dual and $x,x_n\in X$ and $x',x_n'\in X'$, then the following implication holds: $x_n\to_w x$ (weak convergence) and $x_n'\to x'$ in $X'$ $\implies$ ...
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Space of polynomials as a continuous image of F-space

Let $X=\mathbb{R}[a,b]$. Is there any norm $\|\cdot\|$ on $X$ s.th. $X$ is a continuous image of some $F$-space. ($F$-space means that there exists complete metric s.th. $d(x+z,y+z)=d(x,y)$) ? My ...
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Inequality in Banach space [duplicate]

So I have to either prove or disprove this inequality: $$ \left\lVert x\right\rVert^2 - \left\lVert y\right\rVert^2 \le \left\lVert x-y\right\rVert \left\lVert x+y\right\rVert$$ I know this to be ...
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1answer
43 views

infimum of operator norms of iterations of linear operators

I am currently reading a proof in which a fact is used without proof: For a Banach space $X$ and a bounded linear operator $T: X \to X$, $$ \lim_{n \to \infty} \| T^n \|^{\frac{1}{n}} = \inf_{n ...
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2answers
131 views

Weak topology and strong topology in a Banach space.

I have a doubt about weak topology in a Banach space. Let $\mathcal{B}$ a infinite dimensional Banach space, I understood that the weak topology in $\mathcal{B}$, is the topology generated by $\Sigma ...
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38 views

Subspace of a weakly sequentially complete is weakly sequentially complete

A Banach space $X$ is called weakly sequentially complete if all weakly Cauchy sequences are weakly convergent. Question: If $Y$ is a subspace of a Banach space $X$, must $Y$ be weakly sequentially ...
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1answer
41 views

To show a map is an isomorphism

In the proof of Lemma $15$, the author claimed that if there exists constants $C>0$ and $D>0$ such that $$C \sup \{ | \sum_{i=1}^k{\alpha(i) f(x_i) | : \| f \|_{\infty}^1 \leq 1, f(0)=0} \} ...
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46 views

Spectral radius and dense subspace

Let $X$ be a Banach space, and let $E$ be a dense subspace of $X$. Let $A: X \to X$ be a bounded operator on $X$ that maps $E$ to itself. Assume that the spectral radius of $A$ restricted to $E$ is ...
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1answer
37 views

Showing that $(G' \circ u)u' \in L^p(I)$ where $G \in C^1(\mathbb{R})$ and $u \in W^{1,p}(I)$

I am trying to prove the rule of differentiation of a composition for weak derivatives in Sobolev spaces following the proof given in Corolary 8.11 in Functional Analysis, Sobolev Spaces and Partial ...
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1answer
25 views

Why $Y=\{ f \in C^1([0,1]^n) : f(0)=0 \}$ is a closed subspace of codimension $1$

Suppose $C^1([0,1]^n)$ is the set of real-valued functions defined on $[0,1]^n$, whose derivative $\leq 1$ is continuous on $[0,1]^n$. Define $$Y=\{ f \in C^1([0,1]^n) : f(0)=0 \}$$ Why $Y$ is a ...
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Meaning of standard integral convolution

In this paper, in the proof of Lemma $13$, there is this sentence: Now, we find a $1$ Lipschitz $\bar{g} \in C^1(\mathbb{R}^n)$ with $\| f - \bar{g}\|_{|\infty} < \epsilon / 2K$, using the ...
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banach space bigger than $L^p$

we know that $L^p$ is banach space for any $p\geq 1$. My question: Is there any other banach space that is bigger than $L^p$?. In fact, I have an exercice that I don't have any idea: prove that ...
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1answer
36 views

Multilinear Mappings

Let $E$, $F$ complex Banach spaces and $p,q\in \mathbb{N}$ with $p+q\geq 1$. I will denote by $\mathcal{L}_a(^{p,q}E;F)$ the subspace of all $(p+q)$-linear mappings $A\in ...
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1answer
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‎‎‎$‎‎C^*$-algebra generated by ‎$‎‎a$‎

Let ‎$‎‎A$ ‎be a unital ‎‎‎$‎‎C^*$-algebra. ‎‎ Assume that ‎$‎‎a\in A$ ‎is a ‎‎normal ‎and ‎invertible element ‎i.e ‎‎$‎‎aa^*=a^*a$ ‎and ‎‎$‎‎aa^{-1}=a^{-1}a=1$‎.‎ ‎let $‎‎C^*({a}) $ be the ...
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Banach Tarski proof understanding

Theorem (Banach-Tarski Paradox): The unit ball $\mathbb{D}^3 \subset \mathbb{R}^3$ is equi-decomposable to the union of two unit balls. Proof: Let $\mathbb D^3$ be centered at the origin, and $D^3$ ...
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2answers
39 views

Reflexive Banach spaces, compactness

Let $X$ be a reflexive Banach space. Then, consider a linear and compact operator $T \colon X \to X$. Prove that if: $\text{inf} \{ \|Tx\| : x \in X\quad \text{s.t.}\quad \|x\| = 1 \} > 0$, ...
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1answer
45 views

Double dual of the space of bounded operators on Hilbert space [duplicate]

Every Banach space $X$ is canonically, isometrically embedded in its bidual $X^{**}$. But it is not always $1$-complemented in the bidual: for example, there is no projection from $\ell_\infty$ onto ...
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1answer
43 views

Corresponding norm from a dual norm?

Let $(X,N_1)$ be a Banach space (separable if necessary) and let $(X^*,N_1^*)$ be its dual space. Here $N_1^*$ denotes the classical dual norm associated to $N_1$. Let $N_2^*$ be an equivalent norm ...
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1answer
44 views

$C ([1,2] \times [0,1] \to \mathbb R)$ dense in $C ( [1,2] \rightarrow L^{2} ([0,1] \to \mathbb R))$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
2
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1answer
64 views

Pythagorean theorem does require the Cauchy-Schwarz inequality?

In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved ...
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152 views

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$.

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$. Normed spaces are equivalent if there are bounded linear maps which are inverses of ...
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35 views

Looking for an (outside $\Bbb R$) application of a certain theorem

I have the following theorem in the lecture notes: Let $E$ be a normed vector space and $\Omega \subset E$ be open and connected, and let $F$ be a Banach space. Let $(f_n)$ be a sequence of ...
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26 views

On the sequence of orthonormal basic

I have a question : Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such ...