0
votes
0answers
44 views

$C(X)$ is separable when $X$ is compact

Let $X$ be a compact space and let $\Bbb U =\{(U,V); U,V \mbox{ are open subsets of }X \mbox{ and }\mathrm{cl} U \subset V\} $. for $u=(U,V)$ in $\Bbb U$ , let $F_u:X\to [0,1]$ be a continuous ...
-2
votes
0answers
65 views

How to prove this limit in $\ell_1$

Let $w$ be any element in $\ell_1$, and $(w_n)$, $(z_n)$ two bounded sequence in $\ell_1$ that converge to 0 pointwise. Then we have ...
0
votes
0answers
20 views

weakly compact subsets of a Banach space are relative weak topology

Let X be a Banach space and $X^*$ is separable. Show that if K is a weakly compact subset of X, then K with the relative weak topology is metrizable. I can easily show that K with the weak topology ...
2
votes
0answers
44 views

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
0
votes
0answers
32 views

Use of closed and convex set in fixed point property

Let $(X, ||.||)$ be a Banach space and C a subset of X. A mapping $T:C{\to}C$ is non-expansive if $||Tx-Ty||\leq||x-y||$ for all $x, y \in C.$ A Banach space is said to satisfy the fixed point (FPP) ...
1
vote
1answer
30 views

Banach valued sequence spaces $l^p(X)$

Let $X$ be a Banach space and $l^p(X)$ denote the space of sequences $x_i\in X$ for which the norm $\big(\sum_{i=1}^\infty\|x_i\|^p\big)^\frac1p$ is finite, when $X=\mathbb{R}$ we get the usual $l^p$. ...
4
votes
2answers
49 views

Reflexivity of $\ell^p$

I'm having bad difficulties in understanding how to prove that $\ell^p$ with $1<p<\infty$ are reflexive spaces. Every text I have consulted give that as a trivial result because "observing that ...
1
vote
0answers
25 views

Use a fixed point argument to show there exists a unique solution to the following BVP

Show using a fixed point argument that there exists a unique solution $f\in C[0,1]$ to $$ -f''(x)+\sin(f(x))=\sin(x) , x\in (0,1), y(0)=y'(1)=0 $$ This is what I have so far: We can show ...
1
vote
2answers
38 views

A question about reflexive spaces

Quoting wikipedia "a normed vector space is reflexive if it coincides with its bidual". Another definition, more precise is that a normed vector space is reflexive if its evaluation map ...
3
votes
2answers
64 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
71 views

$L^1(μ)$ is finite dimensional if it is reflexive

If $(X,\Omega,\mu)$ is a $\sigma -$ finite measure space, show that if $L^1(X,\Omega,\mu)$ is reflexive then it is finite dimensional. My attempt: I want to show there is a copy of $\ell^1$ in ...
0
votes
0answers
49 views

Proof that a set is open.

Let $(\Lambda_i)_{i\in I}$ a collection of linear operators from $X$ (Banach space) to $Y$ (Normed space). Let $\alpha : X \rightarrow [0,\infty]$ be the function $\alpha(x):=\sup_{i \in I} ...
7
votes
1answer
101 views

Constructing a Banach space of cardinality $\beth_{\omega+1}$

This is related to yesterday's question Constructing a vector space of dimension $\beth_\omega$; it's the next exercise (I.13.35 (a)) in Kunen's Set Theory. Let $B_0 = \ell^1$ and let $B_{n+1} = ...
3
votes
1answer
32 views

Closed subspace of a reflexive Banach space is reflexive

I'm studying Conway's functional Analysis by myself. In page 132 of his book, for showing every Closed subspace M of a reflexive Banach space X is reflexive, he says ...
0
votes
1answer
22 views

Prove that the Besov Space is a Banach space

Help me prove that the Besov space is a Banach space. I need to show that the Besov space is complete. If the Besov space is a closed subset of $L_p$ and since all $L_p$ spaces are complete then I'm ...
0
votes
2answers
51 views

ONB: Density Check?

How to show that $\{\sin{kx}:k\in\mathbb{N}\}$ for $\{f\in\mathcal{L}^2[0,\pi]:f(0)=f(\pi)=0\}$ is an ONB? (Clearly they are orthogonal to each other but is their span also dense?) What general ...
2
votes
1answer
31 views

how do I view the tensor product $X^*\otimes Y$ as a subspace of $\mathcal{L}(X,Y)$?

Background. According to Raymond A. Ryan, in his book Introduction to Tensor Products of Banach Spaces, a Banach ideal $\mathcal{J}$ is an assignment to each pair of Banach spaces $X$ and $Y$ a ...
1
vote
2answers
39 views

Is it a compact operator?

Let $$C^{1}_{2\pi}=\{u\in C^{1}((0,2\pi),\mathbf{R}^n):u(0)=u(2\pi)\}$$ $$C_{2\pi}=\{u\in C^{0}((0,2\pi),\mathbf{R}^n):u(0)=u(2\pi)\}.$$ $C_{2\pi}$ is equipped with the norm $$\|u\|_0=max|u(s)|$$ ...
3
votes
1answer
77 views

How to prove $F$ is a contraction mapping?

Given a $n\times n$ matrix $\mathbf{D}$ in which each entry $\mathbf{D}_{ij} \in [0,1]$. Let $i$-th row vector of $\mathbf{D}$ denote by $\mathbf{D}_{i\ast}$. The mapping $F:[0,1]^n \mapsto [0,1]^n$, ...
0
votes
2answers
60 views

Important applications of the Uniform Boundedness Principle

There's like three applications of the uniform boundedness principle in wikipedia: 1) If a sequence of bounded operators converges pointwise to an operator, then the limit operator is also bounded, ...
2
votes
1answer
49 views

$l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$?

I wonder if $l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$. If so, how to prove it?
2
votes
1answer
39 views

Unconditional bases equivallent to permutations of basis elements.

On page 9 of The Handbook of the Geometry of Banach Spaces: Volume I I found the following: "A basis $\{x_n\}_{n=1}^{\infty}$ is said to be an unconditional basis provided that $\sum \alpha_n x_n$ ...
0
votes
1answer
22 views

Check complex differentiability

I am trying to take a derivative w.r.t $z\in\mathbb{C}$ of the following map: $z\mapsto \sum_{j=0}^{\infty}\lambda_{j} (T(\psi+zh))_{j}$ where $(\lambda_{j})$ is a bounded sequence, $T$ is a ...
1
vote
1answer
21 views

Projections in Tensor Product

Let $X$ and $Y$ be Banach spaces (algebras). If $X\otimes^\pi Y$ denotes the projective tensor product of $X$and $Y$, define $P:X\otimes^\pi Y\rightarrow X$ as follows : for $x\otimes y\in X\otimes ...
1
vote
0answers
65 views

Prove that $L^p+L^r$ is a Banach Space [closed]

all fine? If $1 \leq p < r \leq \infty,$ prove that $L^p + L^r$ is a Banach space with norm $\|f\|=\inf\{\|g\|_p + \|h\|_r ; f = g+h\},$ and if $p<q<r$, the inclusion map $L^q \to L^p + L^r$ ...
2
votes
1answer
38 views

Strict convexity and uniqueness of functionals

Is it true that if $x$ is a norm-one vector in a strictly convex Banach space then there exists a unique bounded linear functional $f$ on that space such that $f(x)=1=\|f\|$? It seems unlikely to me ...
0
votes
2answers
38 views

$T$ closed linear operator, $S \in \operatorname{BL}(B,C)$ invertible implies $ST$ closed

Let $A, B$ and $C$ be Banach spaces, $T: \operatorname{dom}(T)\rightarrow B$ be a closed linear operator with $\operatorname{dom}(T) \subset A$ and let $S \in \operatorname{BL}(B,C)$ be invertible. ...
0
votes
0answers
52 views

a completion of $c_{00}$ under some norm

Let $X$ be the completion of $c_{00}$ under some norm $\|\cdot\|_X$ satisfying $\|e_n\|_X=1$ for all $n$, where we let $(e_n)$ denote the canonical unit vectors in $c_{00}$. In Albiac/Kalton's book ...
4
votes
0answers
42 views

a question about Tsirelson's space

Background. Let $T$ denote the Figiel-Johnson construction of the Tsirelson space, that is, the completion of $c_{00}$ under the implicitly-defined norm ...
2
votes
2answers
27 views

Bounded, surjective linear operator between Banach spaces

How can I show that for a given surjective linear operator $T: X \to Y$ between Banach spaces, if there exists an $\epsilon > 0$ such that $||Tx|| \geq \epsilon||x||$ for all $x \in X$, then $T$ is ...
1
vote
1answer
25 views

$g(T)$ bounded implies $T$ bounded, if $T$ is linear and $g$ is bounded linear functional

Suppose $X$ is a Banach space and $y$ is a normed linear space and $T :X\to Y$ is linear map such that for every bounded linear functional $g\in Y^*$ we have $g( T)$ is bounded . Show that $T$ is ...
2
votes
1answer
21 views

Baire Category for monotonic sequence

Let X be a non-empty complete metric space and let $\{f_n:X\to \Bbb R\}^\infty_{n=1}$ be a sequence of continuous functions with the following property: for each $c\in X$, there exists an integer ...
2
votes
1answer
59 views

In a uniformly convex Banach space $x_n\stackrel{w}\to x$ and $||x_n||\to ||x||$ implies $||x_n-x||\to 0$

In a uniformly convex Banach space $$x_n\stackrel{w}\to x \ \ \text{and} \ \ ||x_n||\to ||x|| \ \ \text{implies} \ ||x_n-x||\to 0.$$ Can you help me to solve it? Thanks in advance.
3
votes
2answers
43 views

Computing the norm of operator when space is equipped with sup norm and $L^1$ norm

Let $\phi $ be the linear functional $\phi (f)=f(0)-\int^1_{-1}f(t)\:\mathrm{d}t$ a.Compute the norm of $\phi$ as a functional on Banach space $C[-1,1]$ with sup norm. b.Compute the of $\phi$ as a ...
0
votes
2answers
42 views

$||\phi||=1$ and $|\phi (x)|=||x||$

a.Let $E$ be a non-zero Banach space and show that for every $x\in E$ there is $\phi \in E^*$ such that $||\phi ||=1$ and $|\phi (x)|=||x||$ b. Let E and F be Banach spaces,let $\pi: E\to F $ be a ...
1
vote
0answers
28 views

Simple tensors in the dual space

Let $X$ and $Y$ be two Banach spaces and assume, if necessary, that $X^*, Y^*$ have the approximation property (but not necessarily the Radon–Nikodym property). Consider the injective tensor product ...
1
vote
1answer
60 views

A question about a dense subset in Banach space.

Let $X,Y$ be Banach spaces, $T:X\to Y$,unbounded linear operator. How to prove that there is a natural number $n$,the set $\{x:\|Tx\|\le n\|x\|\}$ is dense in $X$?
1
vote
1answer
30 views

weak$^∗$ neighborhood of $x$ in $\ell_1$

I have this problem Let $x \in \ell_1$ and $\epsilon>0.$ Choose an $N\in N$ such that $\sum\limits_{k=N}^{\infty}|x_k|<\epsilon$ I cannot understand why V is a weak$^∗$ neighborhood of $x$ in ...
0
votes
0answers
28 views

Frechet derivative and Gateaux derivative

Smulian lemma says Let $(X, ||.||)$ be a Banach space and$(X^*, ||.||^*)$ and let $x\in S_X=\{x\in X:||x||=1\}$ then (i) $||.||$ is Frechet diffrentiable at $x$ iff ...
-1
votes
1answer
60 views

Example of open operator but not closed [closed]

Assume that $T:\ell_1\to\ell_2 $ is bounded,linear and one-to-one. Prove that $T(\ell_1)$ is not closed in $\ell_2$
2
votes
1answer
52 views

the point where all functional are non zero

Let $\{f_n\}$ be sequence of non zero bounded linear functionals on a Banach space X. Show that there is $x\in X$ so that $f_n(x)\ne0$, for all $n\in \Bbb N$. I am confused, non zero functional ...
0
votes
1answer
33 views

Theorems on continuous embedding

Let $X,Y$ be two normed spaces such that $X\hookrightarrow Y$. Show the following: 1) If $A\subset X$ is closed in $Y$ then $A$ is closed in $X$; 2) If $X, Y$ are Banach spaces if $x_n$ tends ...
2
votes
1answer
40 views

Riesz Lemma for reflexive spaces

I know the proof of Riesz Lemma: Let $Y$ be a closed (proper) subspace of a normed space $X$. Let $\varepsilon >0$. Then it exists an element $x \in X$ such that $||x||=1$ and $d(x, Y) \geq ...
1
vote
1answer
57 views

An example of a separable Banach sequence space in which the finite support sequences are not dense?

I am wondering if there exist examples of Banach (or Frechet) sequence spaces in which the set of all finite support sequences are NOT dense?
0
votes
0answers
29 views

Weakly compact closed balls in reflexive space

If it is given that $X$ is a reflexive Banach space. Let $K \subset X$ be a norm closed and norm bounded convex set. I want to show that $K$ is weakly compact. I have the following idea but I am not ...
0
votes
1answer
19 views

Extension of a linear operator

Let $T$ be a linear operator defined on the space of the algebraic polinomials in $[0,1]$ (polinomials with rational coefficients) such that for each $k \in \mathbb{N}, T[x^k]=0$. Is it possibile to ...
0
votes
1answer
41 views

Conicide of $w^*$ and norm topology on $S_{\ell_1}$

I want to show that on $S_{\ell_1}=\{x\in \ell_1: ||x||=1\}$the $w^*$-and the norm topologies are coincide. Can any one help me . Thanks
3
votes
4answers
147 views

Topological Vector Space: $\dim Z\text{ finite}\implies Z\text{ closed}$

Let $V$ be a Hausdorff topological vector space and $Z$ a linear subspace: $Z\leq X$ Is there a neat way to prove that: $$\dim Z\text{ finite}\implies Z\text{ closed}$$
0
votes
0answers
31 views

Reference/confirmation of a result in analysis

Does anyone know of or have a reference for the following result: Let $X$ be a reflexive Banach space with dual $X^{*}$. If there exists a continuous mapping $f: K \rightarrow X^{*}$ on compact ...
3
votes
0answers
104 views

Asymptotically isometric copy of $\ell_1$

A Banach space $X$ is said to contain Asymptotically isometric copy of $\ell_1$ if there is a null sequence$(\rho_n)_{n=1}^{\infty}$ in $(0,1)$ and a sequence$(x_n)_{n=1}^{\infty}$ in $X$ so that ...