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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The restriction of an open bounded linear operator

I need some help with this question. Let $X$ be a Banach space and $T:X \to X$ be a bounded linear operator. Suppose that $T$ is open, and $X_0$ be a closed subspace of $X$. The restriction $T_0$ of ...
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24 views

Show that P is an L-projection iff $P^{*}$ is an M-projection

I have started reading "M-ideals in Banach spaces and Banach algebras", but I stuck on the first page. It says that "there is an obvious duality between L- and M- projections: P is an L-projection ...
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1answer
25 views

Paley projection is not absolutely summing

It is well known that the operator $P: H^{1}(\mathbb{T}) \to \ell_2$, given by $Pf= (\hat{f}(2^{n}))_{n \in \mathbb{N}}$ is bounded and its restriction $P_{|A(\mathbb{D})}: A(\mathbb{D}) \to \ell_2$ ...
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1answer
30 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
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1answer
66 views

A special property of $\limsup$ in $\ell_1$

Let $w$ be any element in $\ell_1$, and $(w_n)$ be a bounded sequence in $\ell_1$ that converge to 0 pointwise. I want to prove ...
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71 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 ...
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27 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
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79 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 ...
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21 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 ...
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88 views
+50

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 ...
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34 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) ...
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3answers
90 views

Differentiation in Banach spaces

Let $E$ be a Banach space, and $F:=L(E,E)$, with $L(E,E)$ the set of continuous linear funtions in $E$. Prove that the function $\exp: F → F$, defined by ...
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1answer
38 views

Banach valued sequence spaces $\ell^p(X)$

Let $X$ be a Banach space and $\ell^p(X)$ 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 ...
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1answer
30 views

Distortion and Norm Stabilization

I'm curious about the Distortion Problem in Banach space theory and its relation with norm stabilization. I found that if $(X, \| \cdot \|)$ is an infinite-dimensional separable Banach space, then $X$ ...
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3answers
45 views

Not a basis for $ l^\infty$ then what is it?

We know that $ l^\infty$ has not a Schauder basis and its Hamel basis is uncountably infinite. Let $e_n=(e_{n1}, e_{n2},...)$ (for each $n\in \mathbb{N}$) s.t. $e_{nj}=0$ when $n\neq j$ and ...
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2answers
51 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 ...
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28 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 ...
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2answers
46 views

The difference between a normed space being reflexive and being isomorphic to its dual

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 ...
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29 views

Sectorial Laplace operator in $L^p$

Can anyone tell me why Laplace operator $\Delta$ is sectorial in $L^p(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is bounded (any other hypothesis are needed?) and $1<p<+\infty$? (When $p=2$ ...
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2answers
68 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) = ...
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40 views

Criteria for being a submanifold

Let $N$ be a subset of a smooth manifold $M$. I want to proof (or a reference for) the following criteria: Every connected component of $N$ is a locally closed submanifold of $M$ if (and only if?) ...
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24 views

Conditional expectation onto maximal abelian subalgebras

If you take a von Neumann algebra $M$ and any its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take ...
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2answers
73 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 ...
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50 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} ...
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102 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} = ...
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1answer
33 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 ...
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1answer
23 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 ...
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2answers
52 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 ...
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Fredholm Integral in Bayesian Appliation

Let $X = x_1, x_2, \ldots, x_n$ be a sequence of Bernoulli random variables with $k$ successes. Suppose that, given $X$, the posterior predictive probability of $x_{n+1} = x$ is known to be $g(x)$ ...
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1answer
32 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 ...
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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 $$\|u\|_0=max|u(s)|$$ ...
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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 $F:[0,1]^n \mapsto [0,1]^n$, ...
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2answers
61 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, ...
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1answer
51 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?
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1answer
40 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 $\sum \alpha_n x_n$ ...
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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 ...
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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 ...
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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
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1answer
39 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 ...
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40 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. ...
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What is the coproduct in the category of Banach spaces and continuous linear maps?

In the category of Banach spaces, where the objects are Banach spaces and the morphisms are continuous linear maps, what are there coproducts? Are they the typical direct sum of Banach spaces? If so, ...
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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 ...
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86 views

Prove that $L^1(\mathbb{N})$ is a Banach space.

I'm trying to prove that $L^1(\mathbb{N}) := \left\{ (x_n)_{n=1}^{\infty} : \sum\limits_{n=1}^{\infty}\left|x_n\right| < \infty \right\} $, the space of all sequences over the field $\mathbb{C}$ ...
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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 ...
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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 ...
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1answer
31 views

$(X/Y)^*$is isometrically isomorphic to $Y^⊥$

Let X be a Banach space with a closed subspace Y, We define the dual mapping $ \pi^*:(X/Y)^* → X^*$ by $\pi^* (\beta)=\beta\circ\pi$ then $(X/Y)^*$is isometrically isomorphic to $$Y^\perp:=\{f∈ ...
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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
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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
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1answer
65 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
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2answers
44 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 ...