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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3
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1answer
37 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
380 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 ?
0
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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 $X$ ? ...
2
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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
77 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 ...
1
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1answer
24 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 $Y$ ?
0
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1answer
32 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 ...
3
votes
2answers
105 views

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 ...
0
votes
0answers
8 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
votes
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
163 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
29 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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0answers
35 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
votes
0answers
27 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. ...
0
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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
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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 ...
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1answer
59 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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0answers
36 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
68 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
27 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
37 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
46 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 ...
1
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1answer
49 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
66 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 ...
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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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23 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.
0
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1answer
41 views

$\{x_n\}$ a sequence in NLS $X$ s.t. $\sum f(x_n)$ converges for all $f \in X^*$ , is the function $f \in X^* \to \sum f(x_n)$ continuous ?

Let $X$ be a NLS , $X^*$ be the set of all bounded real valued functions on $X$ ( i.e. the topological dual of $X$) , let $\{x_n\}$ be a sequence in $X$ such that $\sum_{n=1}^{\infty} f(x_n)$ ...
0
votes
1answer
109 views

Goldstine theorem and weak topology on $E$ induced by weak* topology on $E''$

In the book where I'm studying there this exercise: "(Goldstine's theorem). Recall that $E \subset E''$, or, more precisely, $J_E(E)$ is a subspace of $E''$ where $J_E :E \longrightarrow E''$ is ...
2
votes
1answer
50 views

$X$ be Banach , $T:X \to \mathcal l ^{\infty}$ be linear , $(Tx)_n$ the $n$-th term of $T(x)$;$f_n(x)=(Tx)_n$ ; if each $f_n$ is bdd then so is $T$?

Let $X$ be a Banach space , $T:X \to \mathcal l ^{\infty}$ be a linear transformation , for each $x\in X$ and each $n \in \mathbb N$ , $(Tx)_n$ be the $n$-th term of $T(x)$ and for each $n \in ...
2
votes
1answer
19 views

Proof verification in a functional analysis problem

I am new to Functional Analysis .Please review the following proof: Let $X$ be a Banach space. Let $T:X\to X$ be a invertible linear operator and $M>0$ be such that $\|T^{-k}\|<M$ for all ...
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0answers
16 views

Adjoint lattice homomorphic if surjective

Is the adjoint of a linear operator $T:X\to Y$ between Banach lattices, always lattice homomorphic if $T$ is surjective? This is my proof but I really doubt this is true: $\forall a,b\in Y'$ and ...
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1answer
34 views

Continuity on $L_p$ spaces

Consider a nonlinear and continuos function $f:\mathbb{R} \rightarrow \mathbb{R}$ and we define the functional \begin{equation} F(u) = \int_{[0,1]^2} f(u(x,y)) dxdy \end{equation} where $u$ is an ...
2
votes
1answer
23 views

Are all adjoints lattice homomorphisms?

Obviously something must be wrong in the following reasoning proving that any linear operator $T:X\to Y$ between Banach lattices has a lattice homomorphic adjoint: $\forall a,b\in E':$ $$T'(a\wedge ...
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0answers
54 views

Application of Gronwall Inequality

Let $T>0$ and $f\in C(\mathbb R, L^{2}(\mathbb R))$ with the following property: Put $g(t):= \sup\limits_{0\leq \tau\leq t} \|f(\tau)\|_{X},$ where $X \subset L^{2}$ and $X$ is a Banach Space. ...
1
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1answer
40 views

Sequence of dense sets in Banach spaces

Let $A_0 \supset A_1 \supset A_2 \supset \cdots$ - sequence of embedded Banach spaces and $B_0 \supset B_1 \supset B_2 \supset \cdots$ - suquence of linear spaces such that $B_i$ dense in $A_i$, it is ...
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1answer
34 views

Quotients of $L_1$

I know the rather standard fact in Banach space theory that every separable Banach space is a quotient of $\ell_1$. Is it true that every (possibly non-separable) Banach space is a quotient of some ...
1
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1answer
28 views

Subspaces of quotients of $L^p$ spaces

Is the collection of subspaces of quotients of $L^p$ spaces considered to be a large class of Banach spaces?
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0answers
62 views

Weak-* bounded, closed convex set is compact?

Suppose $E$ is a Banach space, and $K\subseteq E^*$ is convex, and is closed and bounded with respect to weak-* topology. Is it true that $K$ is compact? If $E$ is reflexive, then this is the case, ...
7
votes
1answer
256 views

Holomorphy of a function with values in a Hilbert space

Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2(\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach ...
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0answers
18 views

Unable to find referenced theorem

I've am reading the article "Finitely summable Fredholm modules over higher rank groups and lattices" http://arxiv.org/abs/0806.2759 . Theorem 4.3 here refers to the article Property (T) and rigidity ...
1
vote
1answer
34 views

Locally Lipschitz continuity of the duality map

Let $E$ be a Banach space and let $F(y)$ denote the duality map: $$F(y)=\{y^*\in E^*/ \langle y^*,y\rangle =\|y\|^2=\|y^*\|^2\}$$ where $E^*$ is the dual space of $E$. Are there any sufficient ...
2
votes
1answer
31 views

The limit of a singular matrix?

Now to show the set of invertible $n \times n$ matrices are an open set in the set of all $n \times n$ matrices one can show the set of singular matrices are closed in the set of all $n \times n$ ...
1
vote
1answer
41 views

Real Analysis, Folland problem 5.3.37 Application of the Uniform Boundedness Principle

The Uniform Boundedness Principle - Suppose that $\mathscr{X}$ and $\mathscr{Y}$ are normed vector spaces and $\mathcal{A}$ is a subset of $L(\mathscr{X},\mathscr{Y})$. a.) If ...
4
votes
2answers
65 views

Does every element of the weak-star closure of a set belong to the weak-star closure of a bounded subset?

I feel like this must be a monumentally stupid question. Say $X$ is a Banach space, $S\subset X^*$, and $x^*$ is in the weak* closure of $S$. Must $x^*$ lie in the weak* closure of some norm-bounded ...