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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Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
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the dual space of $L^p$

I am reading some preliminary material to develop a good background in order to study PDE and I came across the following fact The dual space of $L^p$ is $L^q$ where $q$ is the Holder's Conjugate of ...
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Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate.

Hölder's inequality shows that, when $$ \frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$ Is there an example of this ...
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Is the Neumann series a compact operator?

Let $X$ be an infinite dimensional Banach space and $A:X\to X$ be a compact operator with the operator norm $\|A\|<1$. Then $I-A$ is invertible and the Neumann series $$ S_N = \sum_{k=0}^N A^k $$ ...
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Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
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+50

Subsequences of a basic sequence

Suppose ($x_n$) is a basic sequence in a Banach space $X$, and $Y$ is a closed, infinite co-dimensional subspace of $X$. Can we always find a subsequence ($y_n$) of ($x_n$) such that the intersection ...
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20 views

Does Hilbert space with countable dimensions exist? [duplicate]

If there is a Hilbert space with infinite dimensions, can it have countably infinite dimensions? And does Banach space with countable dimensions exist?
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102 views

Proof of equicontinuous and pointwise bounded implies compact

I tried to prove the Arzela-Ascoli theorem: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot ...
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1answer
23 views

Banach spaces: Convergence in terms of the Schauder basis.

Let $X$ be a Banach space. Suppose $X$ has a normalized Schauder basis $\{x_n\}_{n \in \Bbb N}$. Let $\{y_n\}_{n \in \Bbb N}$ be a sequence in $X$ converging to $0_X$. For each $n \in \Bbb N$, let ...
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50 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
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48 views

Norm of a projection in $L_p$

Let $\mu$ be a probability measure on $[0,1]$ and let us define the projection $P$ on $L^p(\mu)$: $$P \; \colon f \mapsto \mathbb{E}(f){\bf 1}$$ (where ${\bf 1}$ is the constant 1 function). What is ...
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27 views

Do stochastic processes form a Banach space?

I'm interested in solving a particular integral equation: $$g(X) = \int_0^1 K(X,p)f(p) \ dp$$ where $f(p)\in L^1([0,1])$ and $X$ is a stochastic process of finite length; i.e. a collection of random ...
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22 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
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1answer
55 views

Why is $I+T$ invertible for this rank-one operator $T$?

I am working with the following lemma given in the book Topics in Banach Spaces Theory: Let ${(x_n)}_{n=1}^{\infty}$ be a basic sequence in Banach Space $X$. Suppose that there exists a linear ...
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32 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
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23 views

How to show that $L^\infty(0,T;L^\infty(\Omega))$ is complete?

How to show that $L^\infty(0,T;L^\infty(\Omega))$ is complete? I did the usual: let $u_n$ be a Cauchy sequence, then we get $$\text{esssup}_t \;\text{esssup}_x |u_n(x,t)-u_m(x,t)| \leq \epsilon$$ Now ...
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1answer
20 views

Unit ball weakly* compact

So here is my problem, I am trying to understand the proof of, $X$ Banach space $\Rightarrow$ the unit ball in $X^*$ is weakly* compact. The proof uses Tychonoffs Theorem to conclude the ...
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31 views

If the quotient of a subspace of a banach space is finite, is it a closed subspace?

Given a Banach space B,V is a subspace of B,if B/V is finite dimension,then is it enough to show that B is closed? Thanks!
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1answer
28 views

Is Limit and Norm interchangable in Banach Spaces

Suppose $X$ is a Banach space, and $\{x_n \}\subset X$. Does it then hold that $\lim \|y-x_n\|=\|y-\lim x_n \| $ ?
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33 views

The image of an bounded and linear operator between two banach spaces is closed if the image is finite codimension?

So here is my problem, I would like to prove the following, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded operator. Then $TX$ is closed if it is of finite codimension i.e ...
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Quotient map, quotient topology in Banach spaces

In Lindenstrauss and Tzafriri's Classical Banach Spaces I an operator $T:X\to Y$ is called a quotient map if the $\overline{TB_X}=B_Y$ where $B_X$ and $B_Y$ are the unit balls in Banach spaces $X$ and ...
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How to get a grip on codimensions

I am trying to find a proof for the following problem: Let $X,Y$ be Banach spaces $A,B:X \rightarrow Y$ are bounded linear operators and $Ran(A)$ is closed. If $\left \| Bx\right \|<\left \| ...
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Confusion with proving that some subspace of a Banach-Space is closed

So here is my problem, I am trying to show that, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded map. Then $T(X)$ is closed if $Y/T(X)$ is of finite dimension. While ...
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81 views

Strongly convex and Frechet differentiable function in reflexive Banach space

We first recall two definitions about strong convexity and Frechet differentiability in normed space. Let $(X, \|.\|)$ be a normed space and $f:X\rightarrow\mathbb{R}$ be a function. (a) $f$ is said ...
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Integration in Banach spaces - interesting, nice and non-trivial examples needed

I am interested in $\textbf{Integration in Banach spaces}$. Here is a little motivation for my question: Let $\left(X,\|\cdot\| \right)$ be a Banach space, $a,b \in \mathbb{R}$ with $a<b$ and $f ...
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1answer
46 views

Spectrum and characters: could anyone please check my proof

I tried to prove the following: Let $A$ be a commutative non-unital complex Banach algebra and $\chi : A \to \mathbb C$ a character. Then $$ \sigma (a) = \{\chi (a) : \chi \in \Omega (A) \} ...
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1answer
56 views

Prove $Tf$ is continuous, $T$ is a contraction and find a solution to the integral $f(x)$

On the Banach space $(C([0,1]), ||.||_\infty)$, consider the operator given by $$(Tf)(x)=\int_0^1 x^2tf(t) dt+1.$$ a.) Prove that $Tf$ is continuous for all $f\in C([0,1])$. b.) Show that $T$ is a ...
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32 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
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On the weak convergence in reflexive Banach space

Consider the following proposition: Proposition 1. Let $X$ be a reflexive Banach space and suppose that $\{x_n\}$ is a sequence in $X$ that is bounded and has at most one weakly sequentially cluster ...
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1answer
53 views

Which are the conditions for a Lorentz space $L^{p,q}$ to be order-continuous?

Which are the conditions for a Lorentz space $L^{p,q}$ to be order-continuous? ( A Banach function space is order-continuous $\equiv$ Increasing sequences of order-bounded positive functions ...
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42 views

The equivalence of norm continuous and weak continuous

Let $X$ be a Banach space and $\phi: X \rightarrow C$ be a bounded linear functional. Then $\phi$ is weakly continuous is equivalent to $\phi$ is norm continuous, right? Why?
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The dual of the Annihilator

Let $X$ be a Banach space, and $I$ be a closed subspace. Then it's known that $(X/I)^*=I^{\perp}$. My question is what is the second dual of $X/I$? or what is the dual of $I^{\perp}$ ? If we know ...
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The space of all bounded sequences over a Banach Algebra.

If $A$ is a commutative Banach algebra, can we identify $l^\infty(A)$ as the dual of $ l_1\hat\otimes A$ (the projective tensor product of $l_1$ with $A$? Also, what do we know about the quotient ...
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1answer
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Continuity of an application between function spaces.

I'm trying to prove the following statement... Let $f:[a,b] \times \mathbb{R} \to \mathbb{R}$ a bounded and continuous function, $t_{0} \in [a,b]$, $x_{0} \in \mathbb{R}$, $r>0$ and $$B= \{ x ...
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Condition to separability of a Banach space.

I am trying to prove the following statement: Let X be a Banach space and $X^{*}$ its topological dual space. If there exists a countable family of functions $(f_{n})_{n} \subset X^{*}$ such that ...
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54 views

Understanding reasons for best constant in inequalities

Why, in functional analysis, is so important to calculate best constant in an embedding inequality? Cross-posted to ...
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Show homeomorphism between convex hull and unit ball?

In the proof of Schauder fixed point theorem in Evans' PDE book, he uses a claim that the convex hull $K$ of $N$ points $x_1,\dots,x_N$ in a convex compact subset $A$ of a Banach space $X$ is ...
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$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
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Need help with understanding the argument in the proof

I am working with one theorem in the book " Topics in Banach Space Theory" by Fernando Albiac and Nigel J.Kalton. I have put a full proof of the theorem here. Here is the book if you want to have a ...
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Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
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Frechet derivative of shift operator in $l_2$?

Let $x \in l_2$ and $J(x) = \sum_{k = 1}^{+\infty} x_k x_{k + 1}$. Find $DJ(u)$ and $D(DJ(u))$. Attempted solution Since $x \in l_2$, then $\sum_{k = 1}^{+\infty}x_k < \infty$. Another fact: ...
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Completeness is not preserved under homeomorphism

I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ($(0,1)$ and $\mathbb{R}$) but I have just thought that ...
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81 views

Mixed Tsirelson Norm

The following is a definition of a Banach space that is a generalization of the original Tsirelson space. Nowadays such a space is called a Mixed Tsirelson space; it was introduced by Argyros and ...
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A problem of weak* continuity in relation with semigroups

Let $(\Omega,\Sigma,\mu)$ be a probability space. Let $\mathcal{A}$ ba a $\sigma$-subalgebra of $\Sigma$. We denote by $\mathbb{E} \colon L^\infty(\Sigma) \to L^\infty(\mathcal{A})$ the associated ...
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69 views

Is the space of continuously differentiable functions over Polish spaces Polish?

Let $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$ be two separable Banach spaces. Consider the space of continuously differentiable functions mapping $X$ to $Y$; i.e. $C^1(X, Y)$. Consider the usual ...
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w*-convergence vs. convergence on a dense subspace

Let us have a Banach space $X$, a dense subspace $D\subseteq X$, a net $\{\phi_{i}\colon i\in\mathcal I\}$ in $X^*$ and $\phi\in X^*$. Suppose that $$\lim\limits_{i\in\mathcal I}\phi_{i}(d)=\phi(d)$$ ...
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32 views

Normed linear space with two norms that are not equivalent, one is complete, what about the other?

I have been searching for an answer to the following question: Given a normed linear space $V$ and two norms that are not equivalent, but $\exists K\in\mathbf{R}$ such that $\|v\|_1\leq K\|v\|_2$ ...
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1answer
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Closure of compact sets in Banach space

Let $(X,\vert\vert\cdot\vert\vert)$ be a Banach space. For each $k\in\mathbb{N}$ let $A_k\subseteq X$ be compact and $r_k\in\mathbb{R},r_k>0$, such that $$A_{k+1}\subseteq \{x+u\vert x\in A_k ...
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1answer
34 views

Dual of Hilbert space dense in dual of Reflexive space.

I don't see how to solve this problem which I think should be easy: Let Y be a reflexive space. Assume $Y$ is continuously embedded in a Hilbert space $H$ and $Y$ is dense in $H$. Show that $H^*$ is ...
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57 views

Uncompletion of a Banach Space

Given any Banach space there is a way to define a norm such that is no longer complete. I know you can reach the result by using a Hamel base $H$ for this given space and doing this: If $$ \forall ...