A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

learn more… | top users | synonyms

0
votes
1answer
55 views

Dense subsets in tensor products of Banach spaces

Assume $B_1$ and $B_2$ are Banach spaces of univariate functions. Moreover, assume that the sets $D_1 \subset B_1$ and $D_2 \subset B_2$ are dense with respect to the respective norms ...
3
votes
2answers
58 views

Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint

Question:Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint. Suppose that the spectrum $\sigma(a)$ is an infinite set. Show that $A$ is infinite-dimensional. How can i prove it? I guess: Let $A$ be ...
1
vote
0answers
33 views

Spectrum of integration operator on $C[0,1]$.

I'm trying to find the spectrum of the operator $T: C[0,1] \to C[0,1]$ given by: $$T(f)(t) = f(0) + \int_0 ^{t} f(s) ds$$ I can show that $0$ is contained in the approximate point spectrum with ...
0
votes
1answer
29 views

Double annihilator of subspace of dual space

If $X$ is a Banach space then it's quite straightforward to show that for $A$ a subspace we have $\bar{A} = {(A^{\circ})}_{\circ}$ and so if $A$ is finite dimensional then $A = {(A^{\circ})}_{\circ}$. ...
0
votes
0answers
15 views

C*−algebras and operator theory, murphy [duplicate]

Do you know how to solve this exercise? (Murphy, $C*$−algebras and operator theory, $2^{nd}$ chapter, 1st exercise) Let $A$ be a Banach algebra such that for all a\in A the implication $Aa=0$ or ...
0
votes
0answers
15 views

Prove map is inflating

Let $T:X\to Y$ be a continuous linear open map between two Banach spaces. Prove that $\exists K\in\mathbb{R}$ such that for each $y\in T(X)$ we have $$T^{-1}(\{y\})\cap B_{K||y||}(0)\neq\varnothing$$ ...
0
votes
0answers
23 views

Second Order Mean Value Inequality In Banach Space

I have some confusions about proving the following theorem from Luenberger's Vector Space Optimization book, Proposition 2 p.176: $\textbf{Claim:}$ Let $X$ be a vector space and Y be a normed space. ...
1
vote
1answer
30 views

A problem which reverses the definition of a bounded operator

I've encontered a problem that appears simple, almost like it's a definition of a bounded operator, but with a reversed inequality sign... and I can't seem to find my way to a solution. Any ...
2
votes
1answer
25 views

example of the arens multiplication; I want to unterstand the construction

I want to understand the double dual as a $C^*$-algebra of a given $C^*-$algebra $A$ but my first problem is to understand the arens multiplication on the double dual $A^{**}$ (considered as Banach ...
2
votes
4answers
64 views

Show that if $\sum x_n$ converges then $x_n \to 0$

Let $(V,\|\|)$be a normed space. Let $(x_n) \subset V^{\Bbb{N}}$. We say that $\sum x_n$ converges if, $\lim_{n\to \infty} \sum_{i=1}^{n}x_i$ exists. Show that if $\sum x_n$ converges then ...
-1
votes
1answer
29 views

${x_n} \to x$ weakly, why does $T{x_n} \to Tx$ weakly? [closed]

If $T \in B(X,Y)$ and ${x_n} \to x$ weakly, why does $T{x_n} \to Tx$ weakly?
0
votes
3answers
66 views

Is a Banach space also a metric space?

Since a Banach space is a complete normed vector space and a norm always induces a metric, a Banach space must be a metric space, right? If so, why is a Banach space defined as a complete normed ...
0
votes
0answers
44 views

Calculating the generator of a weighted transition function

Let $(P_t)_t$ be the transition function of a Feller-Dynkin process $X$. The usual Banach space of functions that the semigroup $(P_t)_t$ is working on is $C_0(E)$, i.e. continuous functions that ...
1
vote
0answers
33 views

Showing projection is continuous if and only if kernel is closed

I have a linear map $P$ on a Banach space, $X$, with $P^2 = P$ and I'm trying to show that $P$ is continuous if and only if $\ker(P)$ and $\ker(I-P)$ are closed. One direction is straight forward but ...
0
votes
1answer
9 views

Determining if linear operator on space of polynomials is bounded

I have $p$ a polynomial given by $p(x) = a_0 + a_1 x + a_2 x^2 ... a_n x^n$ and a linear operator $T$ defined by $T(p)(x) = a_0 + a_1 x^2 + a_2 x^4 + ... + x^{2n}$. The norm on the space is given by ...
-1
votes
1answer
40 views

A question in Banach space. [closed]

Let $C$ be the Banach space of all complex continuous functions on $[0, 1]$, with the supremum norm. Let $B$ be the closed unit ball of $C$. Why there exist continuous linear functionals $\Gamma$ on ...
2
votes
2answers
44 views

Prove $c_0$ is a banach space.

The subspace of null sequences $c_0$ consists of all sequences whose limit is zero. Prove that $c_0$ is a closed subspace of $C$ (The space of convergent sequences), and so again a Banach space. ...
0
votes
1answer
37 views

Predual: Denseness

Problem Given a Banach space $E$. Regard a subspace: $$\iota:U\hookrightarrow E:u\mapsto u$$ Consider the projection: $$\pi:E'\twoheadrightarrow U':\psi\mapsto\psi\circ\iota$$ By Hahn-Banach find: ...
1
vote
1answer
20 views

Differentiability in normed spaces

I really need a help with the following exercise: Suppose $\mathbb{E}$ and $\mathbb{F}$ are normed spaces, $A \subseteq \mathbb{E}$ is an open set, $f: A \to \mathbb{F}$ is differentiable on $A$, and ...
2
votes
1answer
40 views

About closed graph theorem

I want to show that in the closed graph theorem, the completeness of $Y$ is essential. (a.e I want to find two norm space $X,Y$ which $Y$ isn't complete and linear function $T:X\to Y$ such that $T$ is ...
2
votes
0answers
35 views

Characterization of Bochner dual

I want to prove following theorem Let X be separable and reflexive Banach space, $1<p<\infty$ than $$ L^p((0,1),X)^* = L^q((0,1),X^*) $$ where $\frac1{p}+\frac1{q} = 1$, with ...
1
vote
0answers
25 views

Finding isometries of a Banach Spaces.

Given a Hilbert Space $(H,\langle,\rangle)$, $x,y\in H$ and $D\subset H$ a subspace of $H$ (I mean, the operators $+$, $\cdot$ and $\langle,\rangle$ in D are the restrictions of the respective ones in ...
0
votes
0answers
29 views

$L^2-$summand vectors and Paralellogram Law in real Banach spaces

Let $X$ be any real Banach space and $p\in X$, then the Parallelogram Law holds, trivially, for every couple $u,v\in span\{p\}$. We say that $x\in X$ is an $L^2$summand vector of $X$ if ...
2
votes
3answers
47 views

prove that $C_0(X)$ is banach space .

For prove that $C_0(X)$ is banach space X is vector space with norm $||f||_{\infty}$ . I'm trying to prove that $C_0(X)$ is close subset of $C(X)$ therefor i suppose $f \in \overline{C_0(X)}$ so there ...
4
votes
1answer
56 views

How to prove that $(C[a,b], \|\cdot\|_\infty)$ is not a reflexive Banach Space [duplicate]

The tag line basically says it all...this is a question in Luenberger's Optimization book (5.14.4 on p.138). Clearly I don't expect someone to deliver a full proof if it's tedious, but a sketch or ...
0
votes
3answers
39 views

Is this space complete or is it incomplete?

Show (if possible) that the space of all complex sequences $x=(x_n)$ with only a finite number of terms nonzero (the number of nonzero terms may be different for different members of the space) is ...
1
vote
2answers
104 views

Showing that two Banach spaces are homeomorphic when their dimensions are equal.

Let $X$ and $Y$ be Banach spaces. It is quite easy to show that they are homeomorphic when their dimensions are finite and equal. However, I find it difficult to show that they are homeomorphic when ...
1
vote
1answer
36 views

How to prove $(\ell_1\ |.\|)$ is Gateaux differentiable at $x$ if and only if $x_i\not= 0$ for all $i\in\mathbf{N}$?

I am in trouble to prove that $\|x\|=\sum\limits_{i=1}^{\infty}|x_i|$ of $x=(x_i)\in\ell_1$ is Gateaux differentiable at $x$ if and only if $x_i\not= 0$ for all $i\in \mathbf{N}$. I want to use this ...
1
vote
1answer
40 views

A bounded sequence in a Banach space

Let $X$ be a Banach space and $\langle x_n\rangle $ be a sequence in $X$. If ( $f(x_n)$ ) is a bounded sequence for any bounded linear functional $f$ on $X$, then ( $x_n$ ) is a bounded sequence in ...
1
vote
2answers
36 views

“isomorphic” normed spaces and reflexivity

Let X, Y be normed spaces and suppose that there exists an bijective isometry between them. And if X is reflexive, then it is intuitively clear that Y is reflexive also. But, when I tried to prove ...
0
votes
0answers
45 views

Show $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$

Suppose $\{u_n\}$ is a convergent sequence in Hilbert space $H$ and $L$ is a bounded (continuous) linear operator on $H$. Use the definition of convergence to show that $\lim_{n\to\infty}(Lu_n) = ...
1
vote
1answer
54 views

Countable set in a Banach space which spans densely?

Let $\mathcal{C}(\mathbf{T})$ be the algebra of continuous complex functions on the unit circle $\mathbf{T}$. Consider the following two statements: The $*$-subalgebra generated by $1$ and $z$ spans ...
2
votes
1answer
117 views

Proving identity of a $C_0$ semigroup

We have a strongly continuous semigroup $\{T(t)\}_{t\ge 0}$ on a Banach space X with a generator $ A:D(A)\subset X\to X$. I need to show that $\displaystyle T(t)x = \sum\limits_{n=0}^{\infty} ...
4
votes
1answer
32 views

Unit ball separable $\Longrightarrow$ Space separable

Given a normed space $X$ and assume it is also a locally convex space in some other topology (e.g. weak or weak* if it's a dual). Assume that the unit ball $B_X$ is separable in this topology. Is it ...
0
votes
1answer
36 views

Is $B_{\ell_1}$ weak-metrizable?

I know that for a Banach space $X$, the unit ball $B_{X}$ is weak metrizable if and only if $X^*$ is separable. My question is that Is $B_{\ell_1}$ weak-metrizable?
1
vote
3answers
48 views

Showing bounded linear operator has closed image

I'm trying to show that given a bounded linear operator $T: X \to Y$ with $X$ and $Y$ Banach such that $T$ satisfies: For ever $y \in Im(T)$ there is an $x \in X$ with $T(x) = y$ and $||x|| \le ...
1
vote
1answer
38 views

Closed linear operator on Banach spaces.

Let $X$ and $Y$ be Banach spaces and $T : D(T) \rightarrow Y$ a linear operator where $D(T)$ is a linear subspace of $X$. i) Let $T$ be closed and injective. Show that $T^{-1}$ is closed. I tried ...
0
votes
2answers
26 views

A question involving weak and strong convergence

Let E be a Banach space, $K \subset E$ a compact subset in the strong topology and $(x_n)_{n \geq 1} \subset K$, $x_n \rightharpoonup x$ weakly in $\sigma (E, E^*)$. If there is a subseqence ...
4
votes
1answer
32 views

Is $L^1_{loc}(\mathbb{R})$ complete with the norm $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy$

Let $BL^1_{loc}$ be the space of locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy<\infty$. Is this space complete ? What I tried: ...
0
votes
1answer
38 views

Test for normability of a metric on a Banach space

If d is a metric on a (finite dimensional) Banach Space and there exist norms $\|-\|_1$ and $\|-\|_2$ and constants $C_1,C_2 \in [1,\infty)$ satisfying: \begin{equation} C_1\|x-y\|_1 \leq d(x,y) \leq ...
0
votes
0answers
23 views

A question involving duality maps

Show that if X is an infinit dimensional and smooth Banach space, then there are no compact duality maps on X. Can someone, please, give me a hint on how to deduce this from the following fact: Let ...
0
votes
1answer
16 views

It is true that the relatively compact open subsets cannot exist in infinite-dimensional normed spaces?

It is true that the relatively compact open subsets cannot exist in infinite-dimensional normed spaces? Why yes / not? Can someone,please, explain to me? Thank you!
1
vote
1answer
53 views

Extension of bounded operators between norm spaces

Let $X$ and $Y$ be two Banach Spaces and $X_1$ be a subspace of $X$. If $T$ is a bounded linear operator from $X_1\to Y$, then, this is an extension of $T$ from $X\to Y$ such that $\|T\|_{X}\le ...
0
votes
1answer
34 views

Does any continuous map between two Banach spaces over a nonarchimedian field is a closed map?

Does any continuous map between two Banach spaces is a closed map? It seems to me that it is true. Let $f:V \rightarrow W$ be a map, where $V$, $W$ are (infinite dimensional) Banach space over a a ...
4
votes
1answer
36 views

Image of a bounded sequence by a convex continuous function in a Banach space

Let $(X, \Vert \cdot \Vert)$ be a Banach space, and $f : X \longrightarrow \mathbb{R}$ a convex function, continuous for the norm topology. Suppose that $x_n$ is a sequence which weakly converges to ...
4
votes
2answers
123 views

Compute Quotient Space

I have been struggling with this computation for a while now. I thought I was almost there, but it now results I still have nothing. So here is the initial problem: Let $c=\left\{ (x_j)_j \subset ...
3
votes
1answer
46 views

Independent symmetric 3-valued random variables in $L_p$

Consider the following excerpt from this paper: Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...
1
vote
1answer
49 views

Application of the Banach fixed point theorem

Let $a > 0$. We consider the function: $f: (0, \infty) \to (0, \infty)$, defined by $f(x) = \frac{1}{2}(x + \frac{a}{x})$. Let $(x_n)_{n \in \mathbb{N}_0}$ be defined by: $x_0 \in (0, \infty)$, ...
1
vote
1answer
83 views

A copy of $l_\infty$ in a infinite dimensional Banach space

Let $E$ an infinite dimensional Banach space. Using the Hahn-Banach extension theorem, prove that there is a sequence $(y_n)\subset E$ and a decreasing sequence of closed subspaces ...
0
votes
1answer
44 views

The property of closed subspace

We know that a set is closed if and only if every convergent sequence with elements in the set has a limit point in the set. I am reading a paper, and the paper claims that the following is due to S ...