0
votes
2answers
44 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
30 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
36 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
75 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
48 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
36 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
20 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
62 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
36 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
49 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
26 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
23 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
19 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
56 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
42 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
27 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
27 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 [on hold]

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
32 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
56 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
26 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
39 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
146 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
102 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 ...
1
vote
0answers
27 views

Equivalent Frechet differentiable norm on $\ell_1$ and $c_0$

Does there exist an equivalent Frechet differentiable norm on $\ell_1$ and $c_0$? I think we can not find an equivalent norm on $\ell_1$ but we could find an equivalent norm on $c_0$, I do not prove ...
0
votes
1answer
43 views

Frechet differentiable implies reflexive?

Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive? Can any one help me? thanks
0
votes
0answers
34 views

Riemann Sphere: Holomorphic Functional Calculus

Why do we consider the holomorphic functional calculus on the Riemann sphere rather than the complex plane only? Is there a serious problem? Moreover isn't any curve encircling the spectrum ones ...
2
votes
0answers
31 views

Lower semicontinuity of a Bochner integral of a convex function

I'm looking for the following result: Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f$. The map $$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$ is lower semicontinuous for $u \in ...
1
vote
0answers
44 views

Applications of uniformly convex and uniformly smooth of Banach space

I am studying on geometry of Banach space, I want know applications of uniformly convex and uniformly smooth of Banach space in some branches of mathematics and engineering. Can you help me Thanks ...
2
votes
0answers
39 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
0
votes
1answer
30 views

The modulus of smoothness of $c_0$ by an equivalent norm

Let $(X,\|\cdot\|)$ be a Banach space. For $t>0$, the modulus of smoothness of $\|\cdot\|$ is defined by $\rho_X(t)=\sup\left\{\dfrac{\|x+ty\|+\|x−ty\|}{2}−1:x,y\in S_X\right\}$. We define an ...
0
votes
0answers
19 views

Banach-valued holomorphic functions [duplicate]

Let $X$ be a Banach space. Can we define holomorphic functions $f:\mathbb{C}\to X$ by the notion of derivability i.e. $$\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h}$$ Do we still have equivalence between ...
1
vote
1answer
62 views

“right shift” il $L^1$

Let $X=L^1(\mathbb{R})$ be the space of Lebesgue integrable functions $f:\mathbb{R}\rightarrow \mathbb{C}$ with the usual norm. Let $T\in B(X)$ be defined by $$(Tf)(t)= f(t+1)$$ I need to find the ...
2
votes
1answer
42 views

Prove that this space is not Banach

Let $\Omega\subset\mathbb{R}^n$ be an open, bounded set with boundary $\partial\Omega$ of class $C^1$. $$\mathcal{A}:=\{u\in C^2(\bar\Omega):u=0\text{ on }\partial\Omega \}$$ endowed with the scalar ...
0
votes
0answers
29 views

Group of operators such that $|T(t)x|\geq c |x|$

Let $X$ be a Banach space. Can I have an example of a strongly continuous group of operators $T(t)$ such that $$|T(t)x|\geq c |x|, \ t\in\mathbb{R}$$with $c>1$. For $c=1$, I know examples of ...
1
vote
1answer
26 views

Verification of conclusions regarding duality maps

I have two conclusions drawn from two results. I want to know how valid these two conclusions are. Firstly Consider the duality mapping(set-valued) $J:X \rightrightarrows X^{*}$ defined: $J(u) := \{ ...
2
votes
1answer
49 views

Linear Functional: Continuous? [duplicate]

Given a Banach space: $E$ and chosen a Hamel basis: $\mathcal{B}$ Any vector induces a (noncanonical) algebraic linear functional by: $$\delta:E\to E^*:\delta_b(b'):=\delta_{b,b'}\text{ defined ...
2
votes
1answer
50 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
1
vote
0answers
25 views

$\sup_t |T(t)|<+\infty$ implies $\sup_t |T(t)^*|<+\infty$?

Let $X$ be a Banach space. $T(t)$ a family of bounded operators for $t\in\mathbb{R}$. $T(t)^*$ is the adjoint operator of $T(t)$. If $\sup_t |T(t)^*|<+\infty$ , then by Hahn-Banach, there's a ...