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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The closure of the image of the unit sphere

The following seems very credible to me but is it correct? If $E$ and $F$ are Banach spaces, $T:E\to F$ linear and continuous and $\epsilon>0$, $$\overline{TB_E}\subseteq(1+\epsilon)TB_E\,,$$ ...
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31 views

Dense surjection into double dual

Since not all Banach spaces $E$ are reflexive, $$\{(E^\ast\ni f \mapsto f(x)): x\in E\}$$ is not necessarily the whole of $E^{\ast\ast}$. However, is it always dense in $E^{\ast\ast}$?
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36 views

Convergent series which is not absolutely convergent in $(X,\|\cdot\|)$

Let $(X,\|\cdot\|)$ be a normed linear space. Recall from prior results that $(X,\|\cdot\|)$ is Banach $\iff$ any absolutely convergent series in $(X,\|\cdot\|)$ converges. (a) Give an example of ...
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36 views

Continuous inculsion of the dual of continuous included Banach spaces

If $B$ and $C$ are Banach spaces and $B \subset C$ with the inclusion being continuous. If it true that the set of continuous linear functionals on $C$, $C'$, is continuous included in the set of ...
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Contractive Operators on Compact Spaces

Suppose that $T: M \to M$ is a compact contractive Operator on a nonempty compact subset $M$ of a complete metric space $X$. Show that $T$ has a unique fixed point. Further show that the sequence ...
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29 views

Uniform lower bound of sequence of linear maps on a Banach space

Suppose a sequence of $T_j:X\to\mathbb{R}$, with $X$ Banach, has the following property: $$\forall j:\|T_j\|\geq c>0$$ Then we have for all $n$, $$\exists x\in X\setminus \{0\}:\forall j\leq ...
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1answer
147 views

Weak$^*$-convergence of vector-valued measures implies weak$^*$-convergence in $X^*$?

Let $K$ be a compact Hausdorff space and $X$ be a Banach space. By the Riesz-Singer representation theorem, we know that there exists a linear isometry from $C(K,X)^*$ onto $rcabv(K,X^*)$, the Banach ...
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27 views

Norm operators bounded below implies almost uniform lower bound

I have a hard time proving (or disproving) the following statement about continuous linear operators: $$(\exists c>0:\forall j:\|T_j\|\geq c)\Rightarrow(\exists\delta>0:\forall n:\exists x\in ...
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43 views

A question in the proof of $C(X)$ is not a dual space of a Banach space.

Let $X$ be a non-singleton compact connected space.I want to show that $C_{\mathbb{R}}(X)$ is not the dual space of a Banach space,$\mathbb{R}$ is real number field. I already know that, extreme ...
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63 views

Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space

I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces". We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...
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37 views

Degree of infinite dimensinal antipodal map

Let $E$ is a infinite dimensional Banach space and $\Omega=\{x\in E: ||x||=1\}$ . $L:\Omega\rightarrow\Omega,L(x)=-x$, how to compute the degree of $L$? In fact ,I just know that the algebra define ...
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1answer
50 views

On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact?

Suppose that $E$ is a Banach space and let $E^*$ denote its dual space with canonical norm $\lVert\bullet\rVert_{E^*}$. Suppose that $\lvert\bullet\rvert_{E^*}$ is an equivalent norm on $E^*$. The ...
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showing that $f_n(x)= x^n$ is no Cauchy Sequence

Considering the space $C^0([0,1])$ of continuous functions on $[0,1]$, with the norm $||f|| = \max_{x \in [0,1]} |f(x)|$ I have to determine whether $f_n(x) = x^n, n \in \mathbb N$ is a cauchy ...
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1answer
55 views

if Banach space $X\cap Y$ is dense both in $X$ and $Y$, can we have $(X\cap Y)^*=X^*+Y^*$?

Let $X$ and $Y$ be sub-spaces of a large vector space, and both formed Banach spaces with associated norm. If $X\cap Y$ with norm $\|u\|_{X\cap Y}=\|u\|_X+\|u\|_Y$, is dense in $X$ and $Y$ , it's ...
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43 views

Completely continuous map is not homotopy with antipodal map

Define of completely continuous operator : $L$ is continuous operator and map bounded set to relatively compact set , then $L$ is completely continuous operator. Now, $E$ is a infinity dimensional ...
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1answer
36 views

Separation Hahn Banach theorem in vector Banach lattice

Let $X$ be a vector Banach lattice. Let $C$ be a closed cone of positive elements in $X^+$ and let $0\leq x\in X-C$. Q: Does there exists any bounded positive linear functional $f$ on $X$ by which ...
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30 views

Tensor product of $L^1([0,1],\omega)$ with some Banach space

We know that for any Banach space $E$ if $\hat{\otimes}$ denotes projective tensor product then $L^1([0,1])\hat{\otimes} E$ is isomorphism isometric with $$L^1([0,1],E)=\{f:[0,1]\to E:\ \ ...
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1answer
60 views

$x^*\circ f:G\rightarrow \Bbb{C}$ 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
65 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 ...
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1answer
39 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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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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66 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
34 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
42 views

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 ...
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1answer
56 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
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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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50 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 ...
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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 ...
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1answer
60 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 ...
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1answer
70 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
54 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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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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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
106 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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1answer
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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42 views

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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17 views

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 ...
0
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1answer
39 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
128 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 ...
2
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0answers
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 ...