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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5
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A possible norm on a subspace of $C^\infty([0,1])$?

My question is related to this one: Take the vector space of infinitely differentiable functions on $[0,1]$. The standard norm of $C^k([0,1])$ is just the $\ell^1$-norm of the vector $(\|f\|_\infty, ...
0
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0answers
9 views

Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in ...
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1answer
18 views

Does the pth James space Jp contain a norm-1 basic sequence domimated by l2? Equivalently, is there a noncompact operator from l2 into Jp?

The $p$th James space, denoted $J_p$, is just the regular James space using the $p$-norm in place of the 2-norm. See here for a complete definition. To use their notation, let $\mathbb{N}_0$ denote ...
-1
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0answers
17 views

Weak continuity of the duality mapping

Let $X$ be a Banach space, supposed to be reflexif (but not Hilbert), and let $F$ be the duality mapping, supposed to be univoque and Lipschitz. I'm looking for a sufficient condition under which ...
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52 views

Functional Analysis Homework [on hold]

It is half of my homework. I can not do it myself. Please help me..
0
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1answer
30 views

weak convergence and weak * convergence criterium

I have to solve the following problem Let $X$ be a Banach space. Prove that $x_n\rightharpoonup x$ in $X$ if and only if $\sup||x_n||<+\infty$ there exists a dense subset $E'$ of ...
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0answers
21 views

A confusion about the norm of the restriction of a linear mapping.

Let $\Bbb X$ be a Banach space, $T:\Bbb X\to \Bbb X$ be a linear map and $P:\Bbb X\to \Bbb X$ be a projection operator. Denote the closed subspace that is the range of $P$ by $\Bbb Y:=\mathcal R(P)$. ...
0
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1answer
29 views

Finite Dimensional Hilbert Space

A while ago someone asked this question. I really like what the accepted answer is trying to do. But, I am having trouble figuring out his justification for the first line in the proof: $$\bigcup_{x ...
0
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2answers
47 views

Why does this proof work: Closed unit ball in $C_0$ is not compact

I know that this question has been asked to death, and multiple solutions are given, but I still don't understand why the "standard" proof works Following Show that the closed unit ball $B[0,1]$ in ...
0
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1answer
28 views

Hahn Banach Theorem Application

I want to proof that exists $f \in l_\infty '$ with $f(x) = \lim x_n, \forall x = (x_n) \in c$ and $f(x_1, x_2, x_3,...) = f(x_2,x_3,x_4,...)$ What I have been doing until now: Consider the ...
0
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1answer
35 views

About a particular linear map between sequence spaces

Let $x \in \ell^1$ and $z \in \ell^2$ taking values in $\mathbb{R}$ and define a linear map $T_z: \ell^1 \rightarrow \ell^2$ as follows: $y_1=0$ and $y_n=\sum_{k=1}^{n-1}z_{n-k}x_k$ for $n\geq 2$. ...
1
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1answer
37 views

Weakly convergent series

Consider sequence $\{x_n\}_{n = 0}^\infty $ in complex Banach space X. Suppose that for all $z \in \mathbb{C}$ such that $|z| < 1$ the series $\sum_{n = 0}^\infty z^nx_n$ is weakly convergent. How ...
2
votes
1answer
44 views

Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
3
votes
1answer
75 views

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions?

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions? My attempt to prove this: For contradiction, suppose $X$ is an ...
2
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2answers
36 views

Sequence of bounded Operators (Is this a counterexample?)

I've to proof the following statement Let $X,Y$ be to banach spaces and $(T_k)_{k \in \mathbb{N}} \subseteq L(X,Y)$ a bounded sequence of bounded linear operators. Further it exists a dense subset ...
0
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0answers
23 views

Banach space and invertible linear operator

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
0
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1answer
37 views

Proof, that a set is not convex

I try to solve the problem 106 of the scottich book. I know the set of all rearranged sums is convex. Let $f_{j,k}$ the indicator function of the interval $(\frac{j}{2^k},\frac{j+1}{2^k}$). k = ...
1
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1answer
39 views

Symmetric norms on $\mathbb{R}^n$ are between the $\ell^1$ and $\ell^\infty$ norms

We say that a norm $\|\cdot\|$ on $\mathbb{R}^n$ is symmetric if it is invariant under sign changes and permutations of the components. Suppose that $$\|(1, 0, ..., 0)\| = 1$$ for a symmetric norm. ...
3
votes
1answer
35 views

completion of $C^{\infty}\left(S\right)$ is $L^2(S)$?

I have the space of infinitely derivable functions, i.e. $C^{\infty}\left(S\right)$ with the following inner product $$\left\langle f,g\right\rangle ...
3
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2answers
370 views

Can the real vector space of all real sequences be normed so that it is complete ?

Let $X$ be the vector space of all real sequences . Does there exist a norm on $X$ which makes it complete ?
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1answer
19 views

Affine contractions from linear contractions?

Let $V$ be a linear space. Consider a contractive linear map $M:V\mapsto V$, $$ \|Mv\|\leq \|v\| \quad \text{for all vectors } v\in V. $$ Now, for some fixed vector $c\in V$, the question is to sort ...
0
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2answers
21 views

$X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?

Let $X$ be a finite dimensional real normed linear space , let $x \in X$ , then does there exist a continuous linear transformation $T:X \to X$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$ ? ...
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0answers
20 views

$f \in \mathcal l^{\infty}{'} $ ; $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence of non-negative terms ; is $f$ bounded? [duplicate]

Let $f:\mathcal l^{\infty} \to \mathbb R$ be a linear functional such that $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence with non-negative terms ; then is $f$ continuous ?
5
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1answer
72 views

isomorphism between $C[0,1]$ and $C^1[0,1]$

Is space $C[0,1]$ with norm $\parallel f \parallel=\max|f(x)|$ (space of continuous functions on $[0,1]$) isomorphic to space $C^1[0,1]$ with norm $\parallel f \parallel=\max|f(x)|+\max|f'(x)|$ (space ...
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1answer
14 views

A complex unital algebra which is a Banach space is also a Banach algebra

In my functional analysis class I got stuck on this question on Banach algebras: Let $ \mathbb{A} $ be a complex unital algebra (it has a unit i.e. a multiplicative identity) and $ \mathbb{A} $ is ...
0
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1answer
15 views

$X$ be Banach space with a proper dense subspace $Y$. Can the identity operator on $Y$ be extended to a continuous function from $X$ into $Y$?

Let $X$ be any Banach space with a proper dense subspace $Y$. Can the identity operator on $Y$ be extended to a continuous function from $X$ into $Y$ ?
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1answer
25 views

Problem in functional analysis: application of open mapping theorem

I'm having trouble in exercise 2.10 in Brezis' book in Functional Analysis: let $E$ and $F$ be two Banach spaces and let $T \in \cal{L}$$(E,F)$ be surjective. Let $M$ be any subset of $E$. Prove that ...
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2answers
93 views
+100

Show $F_b (\Omega, X)$ is a Banach space

Let $\Omega$ be any non empty set and let $X$ be a Banach space over $\mathbb{C}$. Let $F_b (\Omega,X)$ be a linear subspace of $F(\Omega, X)$ of all functions $f; \Omega \to X$ such that ...
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0answers
7 views

Tensor product of algebras of operators on $SQ_p$ spaces

I am aware that there is a canonical tensor product for $L_p$ spaces that allows one to talk about tensor products of algebras of bounded linear operators on $L_p$ spaces. Is there an analogous notion ...
2
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0answers
51 views

Is the set of adjoint operators weak* closed?

Suppose we have a Banach space $X$ and a net of bounded operators $(T_\gamma)$ on $X$ such that $T_\gamma^*\to S$, for some bounded operator $S$ on $X^*$, where the convergence is with respect to the ...
2
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3answers
132 views

Weak convergence $\iff$ strong convergence in finite dimensional space

I am seeking a proof of the following claim. Weak convergence $\implies$ strong convergence in a finite-dimensional normed linear space. Thank you.
2
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1answer
40 views

Spectrum of an Operator on a Banachspace

Claim: Let $A$ be a bounded linear operator on a Banachspace $\mathfrak{X}$. Denote $\sigma(A)$ as the spectrum of A. Let $\lambda$ be a point in the boundary of the $\sigma(A)$. Then there exist a ...
0
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1answer
24 views

$X,Y$ be Banach , $T \in \mathcal B(X,Y)$ be onto ; then , for every sequence $y_n \to y \in Y$ , $\exists x_n \to x\in X$ s.t. $T(x_n)=y_n , T(x)=y$?

Let $X,Y$ be Banach spaces , $T:X \to Y$ be a surjective continuous linear transformation , then is it true that for every convergent sequence $\{y_n\}$ in $Y$ , converging to $y \in Y$ , there exist ...
0
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0answers
39 views

$Y,Z$ be linear subspaces of a Banach space $X$ ; $Y$ be finite dimensional , $Z$ closed in $X$ ; is $Y+Z$ closed in $X$? [duplicate]

Let $Y$ and $Z$ be linear subspaces of a Banach space $X$ , such that $ Y$ is finite-dimensional and $Z$ is closed in $X$ , then is $Y +Z$ also closed in $X$ ?
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31 views

Continuity in dual space with weak$^*$- topology

Let $X$,$Y$ be locally convex topological vector spaces. Assume now I have an operator $T:Y'\rightarrow X'$ where $Y'$ and $X'$ are equipped with the weak$^*$-topology. Does this imply that $T$ is ...
4
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0answers
24 views

References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. ...
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0answers
20 views

$C[a,b]$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space [duplicate]

Let $-\infty\lt a\lt b \lt \infty$. Let $C[a,b]$ is space of continuous functions and a function $\|.\|:C[a,b]\to R$ is given by $\|f\|:=\int_a^b |f(x)|d(x)$ a). Show that $\|\cdot\|$ is a norm in ...
4
votes
1answer
38 views

Differentiability of an action of the group of invertible elements of a $C^{*}$-algebra $\mathcal{A}$ on the dual of $\mathcal{A}$

I am studying the actions of Banach-Lie groups on Banach manifolds, and I am not able to concretely evaluate the differentiability properties of a specific action. Let $\mathcal{A}$ be a unital ...
-1
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1answer
54 views

The space $C[a,b]$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space

Let $-\infty\lt a\lt b \lt \infty$. Let $C[a,b]$ is space of continuous functions and a function $\|.\|:C[a,b]\to R$ is given by $\|f\|:=\int_a^b |f(x)|d(x)$ a). Show that $\|\cdot\|$ is a norm in ...
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0answers
19 views

Extreme positive functionals on $\ell^\infty$

Let $\phi$ be a positive functional on $\ell^\infty$ such that $\phi((1,1,1,\dots))=1$ which cannot be written as a non-trivial convex combination of such functionals. Is $\phi$ necessarily ...
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31 views

Do linear transformations of convergent sequences converge?

Let $T \in \mathcal{L}(E,F)$ where $E, F$ are two normed vector spaces (not necessarily finite or complete (Banach)). Is it true that if $x_{n} \rightarrow x$ in $E$, is it true that $Tx_{n} ...
2
votes
1answer
48 views

$\|L(v)\| \leq \|L\|\cdot\|v\|$ on Banach spaces

Let $A,N$ be Banach spaces and let $L: A \rightarrow N$ be a linear transformation. If $L$ is continuous (which is guaranteed on finite dimensional spaces), the set $ \{ M \geq 0 \ : \|L(v)\| \leq ...
1
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1answer
67 views

The space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ [closed]

Is the space of continuous function from the set of the number $\mathbb{R}$ to $\mathbb{R}$ (usually denoted by $C(\mathbb{R},\mathbb{R})$ is a Banach space? With the norm $\mathop {\sup }\limits_{x ...
0
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1answer
26 views

Pseudo-resolvent function

Let $\emptyset \neq D$ a open set in $\mathbb{C}$ and $J: D \to B(E)$ a continuos function such that $J(\lambda) - J(\mu) = (\mu - \lambda)J(\lambda) J(\mu)$ where $E$ is Banach space. We must show ...
2
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1answer
36 views

Show that the closure of $C_c(X)$ is $C_0(X)$.

Let $(X,T)$ be a topological Hausdorff space. By $C_b(X)$ denote the continuous bounded function $f\colon X\to\mathbb{R}$, by $C_c(X)$ the continuous functions $f\colon X\to\mathbb{R}$ which have ...
1
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1answer
43 views

Show there is a continuous isomorphism $l_{\infty}\rightarrow \left(l_1\right)^* $

Let $\left(l_1\right)^*$ be the dual space to $l_1$. Each $f \in \left(l_1\right)^*$ is a continuous linear functional over $l_1$. There is constant $C \in \Bbb R$ such that $|f(x)|\le C|x|_1, \forall ...
0
votes
1answer
40 views

Show, that $c$ and $c_0$ is a Banach space

Let $c=\lbrace x=\lbrace x_n\rbrace ,n\in \mathbb{N}: \exists \, \text{lim}_{n\to \infty}x_n\rbrace$ and $c_0=\lbrace x=\lbrace x_n\rbrace ,n\in \mathbb{N}: \text{lim}_{n\to \infty}x_n=0\rbrace$. I ...
3
votes
1answer
61 views

Are the weak* and the sequential weak* closures the same?

I have a question that might be easy for the experts. Let $E$ be a Banach space (non-separable) and let $E'$ be its dual space. Suppose that $X\subset E'$ and assume that $X$ is separable with ...
1
vote
0answers
17 views

Proving Banach space is finite in Baire space [duplicate]

Let $B$ be a Banach space where the dimension of the underlying vector space is countable. using the Baire category Theorem, prove that the dimension of the underlying vector space is, in fact, ...
0
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0answers
22 views

On bounded bijective linear maps

If $f$ is a bounded bijective linear map from a Banach space $E$ to $E$. How can one prove that if $(f(x_n))$ converges to $0$ then $(x_n)$ also converges to $0$ without using the open map theorem.