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

3
votes
0answers
11 views

Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous? If so, is it also smooth? Does it induce a continuous (smooth) map ...
0
votes
0answers
19 views

About isometry on Banach spaces

Let $X$ be a Banach space. For every $x\in X,$ the non-empty dual duality set $\mathcal{J}(x)$ is defined as:$$\mathcal{J}(x):= \left\{j(x) \in X': \langle x, j(x)\rangle = \|x\|^{2} = \|j(x)\|^{2} ...
1
vote
1answer
17 views

How to show $\sigma(T_q) = \overline{\{q(t) : t \in [0,1]\}}$ where $T_q$ is the multiplication operator?

Let $B$ be the Banach space of bounded complex functions on $[0,1]$ with sup-norm. For $q \in B$, define the (multiplication) operator $T_q : B\rightarrow B$ by $(M_q f)(t) = q(t)f(t)$. How do you ...
1
vote
2answers
33 views

What is Convex about Locally Convex Spaces?

This might be a silly question, but what motivates the name "locally convex" for locally convex spaces? The definition in terms of semi-norms seems to have nothing to do with convexity or with the ...
3
votes
1answer
112 views
+50

Show that $(L^{p},\|\|_{p})$ is a Banach space.

Show that $(L^{p},\|\|_{p})$ is a Banach space. My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, ...
2
votes
2answers
392 views

Second derivative of a composite function

Say, we have three Banach spaces $X, Y, Z$ and $g:X \to Y, \ \ f:Y \to Z$ are twice (Fréchet) differenciable. The question is: what is $(f \circ g)''$? Since $(f \circ g)'':X \to ...
3
votes
1answer
66 views

Is $C^\omega([0,1])$ normable? (And about the growth of coefficients of infinitely differentiable functions)

This question arised to me when trying to prove that the space of infinitely differentiable functions defined in a compact space $K\subset\mathbb{C}$ taking values in $\mathbb{C}$, that is ...
2
votes
2answers
498 views

Nearest point projection in uniformly convex Banach spaces

Let $X$ be a uniformly convex Banach space, $x\in X$ and $C\subset X$ closed and convex, then there is a unique $y\in C$ with $$\lVert x-y\rVert=\inf_{z\in C}\lVert x-z \rVert.$$ Is there a good book ...
1
vote
1answer
31 views

Question related to differentiable functions on Banach spaces

There is an interesting exercise on my Analysis book that I have not been able to solve: Let $\mathbb{E,F}$ be Banach spaces, $f:\mathbb{E}\to\mathbb{F}$ of type $\mathcal{C}^k$, $k\geq1$. Asume ...
6
votes
1answer
73 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm ...
0
votes
1answer
34 views

Continuity on [0,2) of the modulus of convexity

For a Banach space $X$ the mapping $$\delta_X\colon[0,2]\to[0,1], \varepsilon\mapsto \inf\{1-|x+y|/2\colon |x|\leq 1, |y|\leq 1, \text{and } |x-y|\geq\varepsilon\} $$ is called the modulus of ...
2
votes
0answers
32 views

am I misunderstanding boundness of a transformation?

Let $X$ be a Banach's space $Y$ a normed vector space $H\in B(X,Y)$ a family of bounded linear transformations $X\rightarrow Y$ and $V_n:=\{x\in X:\exists T\in H$ such that $||Tx||>n \} n\in ...
3
votes
1answer
35 views

Sum of closed spaces is not closed

I recently encountered the theorem that the sum of a closed (linear) subspace with a finite dimensional subspace is closed subspace of the Banach space in which it is contained. However, this came ...
4
votes
1answer
29 views

Sequences - Convolution

We got the following setting: Let $A$ the set of all complex sequences $x=(x_n)_{n \in \mathbb N_0}$ with \begin{align*} \Vert x \Vert := \sum_{n = 0}^\infty \vert x_n \vert e^{-n^2} < \infty. ...
3
votes
1answer
177 views

Dual space is 1-complemented in the third dual

Given $X,Y$ be normed spaces over $\mathbb k$ (where $\mathbb k$ could be $=\mathbb C$ or $\mathbb R$). Let $T\in L(X,Y) $ (a continuous and linear operator) then we define $T':Y'\to X'$ by ...
0
votes
2answers
16 views

$\bigcap_{S \in L(E,F)} ker(S) = \{0\}$

Let $E$ a Banach space, $F$ a normed space and $L(E,F)$ a set of bounded linear operator from E to $F$. Is true that $$\bigcap_{S \in L(E,F)} ker(S) = \bigcap_{S \in L(E,F)} S^{-1}(0) = \{0\}.$$ If ...
0
votes
0answers
21 views

Interpolation of a subspace of codimension one

I am a little bit lost in interpolation theory. Let $A$ be a linear operator on $\mathbb{R}^{n}.$ Denote $V_0$ a subspace of $\mathbb{R}^n$ of codimension one. Suppose that $AV_0 \subseteq V_0$ and we ...
2
votes
2answers
39 views

Schauder bases of subspaces of the sequence space $\ell^p(\mathbb{N})$

Consider the canonical Schauder basis $\{e_i:i\in \mathbb{N}\}$ for $\ell^p(\mathbb{N})$, where $e_i(j)=\delta_{ij}$. Let $M$ be a subspace of $\ell^p(\mathbb{N})$. Is it right that $\{e_i:i\in ...
0
votes
1answer
22 views

The space $C_0$ does not have a 2-dimensional subspace isometric to $E^2$?

To prove that the space $C_0$ of the sequences x=($x_1,x_2$,...) for which $\displaystyle \lim_{i=\infty} x_i=0$ with the norm $\|x\|=\max_{1 \leq i < \infty} |x_i|$ does not have a two-dimensional ...
6
votes
1answer
225 views

Banach Spaces: Uniform Integral vs. Riemann Integral

Problem Given a finite measure space $\Omega$ and a Banach space $E$. One has strict inclusion: ...
0
votes
2answers
30 views

counterexample for $C^1(U)$ not complete in any dimension

Cleary $C^1[a,b]$ is not complete with $\|\cdot\|_{\sup}$. I am looking for a counterexample which is working in any dimension, i.e. $C^1(U)$ is not complete for any open $U\subseteq \mathbb R^n$ ...
0
votes
0answers
48 views

Weak closure of the boundary of the unit ball of $l^1$

I am trying to prove that zero is in the weak closure of set $\{ x\in l^1 \,:\, \lVert x\rVert =1\}$ . And I need example too. Is there any countable subset of this set which has $0$ in weak closure?
0
votes
0answers
26 views

Convergence of series of linear operators in Banach space

Let $X$ be a Banach space and $A\in L(X,X)$. Show that $$\sum_{k=0}^\infty \frac{A^k}{k!} $$ converges in $L(X,X)$. Find an upper bound to the norm of sum. If a series converges absolutely, then it ...
3
votes
1answer
51 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 ...
1
vote
1answer
27 views

Existence (and construction) of a convergent series

Suppose $(b_n)$ is an unbounded (real or complex) sequence. Does there always exist some (absolutely) convergent series $\sum a_n$ such that $\sum |a_nb_n|$ (or better, $\sum a_nb_n$) diverges? If so, ...
1
vote
0answers
26 views

the “real spectrum” of an operator acting on a real Banach space

Let $X$ be a Banach space over the field $\mathbb{R}$, and denote by $\mathcal{L}(X)$ the space of continuous linear operators acting on $X$. The spectrum $\sigma(T)$ of an operator ...
17
votes
1answer
406 views

Are there spaces “smaller” than $c_0$ whose dual is $\ell^1$?

It is well known that both the sequence spaces $c$ and $c_0$ have duals which are isometrically isomorphic to $\ell^1$. Now, $c_0$ is a subspace of $c$. My question - is there an even smaller subspace ...
0
votes
1answer
23 views

Show that there is a unique function on the interval that solves:

$$f(x) = e^{-2x} + \int_0^\infty e^{-2x-2y} \sin(x-y)f(y) \, dy$$ I can't get a good bound on $e^{-2x-2y} \sin(x-y)$ so that I can apply Banach.
3
votes
1answer
49 views

spectrum of a unilateral left weighted shift operator on c0

The spectrum of shift operators on $\ell_p$ is well-documented, however the spectrum of shift operators on $c_0$ does not seem to be nearly as commonly discussed. Definitions. Let ...
2
votes
2answers
104 views

A uniformly continuous on the unit sphere of $c_0$ is bounded

Let $S_{c_0}= \{ x\in c_0 \;: \Vert x \Vert =\;1\; \}$, where $c_0$ is the space of all sequences converging to zero and $$f: S_{c_0} \rightarrow \mathbb R$$ is a uniformly continuous function. Prove ...
1
vote
1answer
57 views

Not unique Hahn Banach extension

$G = \{(x_n) \in l_1: x_{2n+1} = 0, \forall n \in \mathbb{N} \}$ Let $f: G \to \mathbb{K}$ be a continuous linear functional, $f \neq 0$. Show that the Hahn Banach extension of $f$ is not unique. ...
7
votes
1answer
65 views

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, ...
1
vote
1answer
27 views

Uniqueness of a Hahn Banach extension

Let $M = \{(x_n) \in l_1: x_1-3x_2=0\}$ and $f: M \to \mathbb{R}$ given by $f((x_n)) = x_1$. Let $g: l_1 \to \mathbb{R}$ be the Hahn Banach extension of $f$. Show that $g$ is unique. I know that ...
0
votes
1answer
42 views

Problem. Convergence. Banach space. Weak topology

Let E be a Banach space and let $(X_n)$ be a sequence such that $X_n \rightharpoonup x$ in the weak topology σ(E,E'). Set: $S_n=\frac{1}{n}\sum_{k=1}^n(-1)^kX_k$ Does $Sn \rightharpoonup x $ in the ...
0
votes
0answers
32 views

Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
0
votes
0answers
43 views

A not so obvious corollary?

Let $X$ be a normed vector space over $\mathbb{K}$. Then there is only one completion of $X$, the banach space $\hat{X}$ such that $X$ is a dense subspace of $\hat{X}$. I am trying to prove that there ...
6
votes
3answers
2k views

Dual of $l^\infty$ is not $l^1$

I know that the dual space of $l^\infty$ is not $l^1$, but I didn't understand the reason. Could you give me a example of an $x \in l^1$ such that if $y \in l^\infty$, then $ f_x(y) = ...
3
votes
2answers
31 views

Isometric embedding of $\ell ^ 1$ in $\ell ^\infty$ in finite dimensions

It is well known that $\mathbb{R}^n$ with $\ell ^1$ norm can be embedded into $\mathbb{R}^k$ with $\ell^\infty$ norm for some $k\in \mathbb{N}.$ But I guess, this is not true in complex case that is ...
0
votes
0answers
19 views

Banach Theorem on Metric Space for Integral Equations

My instructor said that Banach doesn't apply in this case: f(x) = sin(x) + $\int_0^x$$f^2$(z)dz f(0) = 0; f'(0) = 1 > 0 f'(x) = cos(x) +$f(x)^2$, which is positive on (0,$\pi$) so f is positive ...
0
votes
1answer
61 views

Prove that $(a_n) \in l_2$.

Suppose that $\sum_{k=1}^{\infty} a_k x_k$ converges for all $x=(x_n)\in l_2$. Prove that $(a_n) \in l_2$. My attempt Let $T_n: l_2 \to \mathbb{K}$, $T_n(x) = \sum_{k=1}^{n} a_k x_k$. ($\mathbb{K} ...
0
votes
0answers
37 views

Is there a Schauder basis of the Banach space $l_p(l_q)$

For Banach space $l_p(l_q)$, where $p,q\geq 1$, does the following vectors form a Schauder basis of it, say a rearrangement of all $v_{m,n}=(0,...,0,e_m,0,...)$, where $e_m$ occur in $n$th coordinate ...
4
votes
0answers
31 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 ...
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 ...
1
vote
1answer
41 views

Norm of linear operator

Given two real numbers $\alpha$ and $\beta$, consider the linear operator $T:\mathbb{C}\rightarrow \mathbb{R}$ defined by $T(x+iy)=\alpha x +\beta y$. I am trying to figure out the norm of this ...
2
votes
1answer
134 views

When is the dual ball of $L_1(\mu)$ weak*-sequentially compact?

Where could I find a direct proof showing that the dual ball of $L_1(\mu)$ is weak*-sequentially compact? Since $(L_1(\mu))^*=L_\infty(\mu)$, I mean the unit ball $B_{L_\infty(\mu)}$ of ...
2
votes
1answer
38 views

example of the arens multiplication; I want to understand 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 ...
5
votes
0answers
73 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
1
vote
3answers
261 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
1
vote
0answers
69 views

$L^{\infty}$ is p-concave

I want to show that the $L^{\infty}$ on a Banach lattice $X$ is $p$-concave with $M_{(p)}(L^{\infty})=1$. Where $L^\infty=L^\infty(X,\mathcal{M},\mu)$. Recall that a Banach lattice $X$ is said ...
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. ...