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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Image of a set under a mapping

I need to show that the image of the closed unit ball in $\mathbb{C}$, under the polynomial mapping $p(x) = (1-x)^2$ is the cardioid: ${re^{i\theta} : 0 \leq \theta < 2π, 0 ≤ r ≤ 2 + 2 \cos\theta}$...
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Please help me proving that the sum is harmonic

If ${c_n}$ is a bounded sequence, then $$ f(r, \theta)=\sum_{n=-\infty}^{\infty}c_nr^{|n|}e^{in\theta} $$ is harmonic in the disc. Help me proving?
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314 views

separable Banach space with Banach-Mazur distances to $\ell_2^n$ bounded must be isomorphic to $\ell_2$?

If $X$ is a separable infinite-dimensional Banach space and $C\in\mathbb{R}^+$ is an upper bound for the Banach-Mazur distance $d(E,\ell_2^n)$ for all $n\in\mathbb{N}$ and all $n$-dimensional $E\leq X$...
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1answer
59 views

existence of invertible operator mapping one sequence pointwise to a 'nearby' sequence

Let $X$ be a Banach space and $(x_n)$, $(y_n)$, $(f_n)$ be bounded sequences in $X$, $X$, $X^*$ respectively such that $f_m(x_n)=\delta_{mn}$ $\forall m,n$ and $\epsilon=\Sigma\|x_n-y_n\|<\infty$. ...
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364 views

A question about Banach limits

I'm trying to prove that there exists a multiplicative linear functional in $\ell_\infty^*$ that extends the limit funcional that is defined in $c$ (i.e., im looking for a linear functional $f \colon \...
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1answer
400 views

Strong and weak-* convergence for bounded linear maps

The textbook I am using says that a sequence $(T_n)$ in $\mathcal{B}(X,Y)$ for $X$, $Y$ normed linear spaces converges strongly to $T$ if $\lim_{n\rightarrow\infty}T_nx=Tx$ for every $x\in X$. The (...
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1answer
198 views

Banach Space continuous function

On the Banach space $(C([-1,1]), ||\cdot||_\infty ) $ consider the operator given by $(Tf)(x)= \dfrac{1}{3} \displaystyle\int^1_0txf(t)\ dt + e^x - \dfrac{\pi}{3} $ 1) prove that the mapping is a ...
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255 views

Closed range operators

Let $T$ be a linear operator between two normed spaces. I'm trying to show that an operator $T$ has closed range if and only if $\operatorname{im}(T) = (\ker{(T^*)})^{\perp}$. Is there a way to do it ...
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1answer
180 views

Dual spaces of complex sequences, show the second member is in the dual space

I'm having trouble with some of (ok, most of) the exercises in my 1st-year-master's functional analysis class, so here's one of them, hoping someone can help me out: If a sequence $(b_n)$ is ...
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1answer
140 views

basic sequence and strictly singular operators

Suppose that $T: X\to Y$ is strictly singular. Prove 1) In every infinite dimensional subspace $Z$ of $X$, there exists a normalized basic sequence $(x_n)$ such that $||Tx_n|| <2^{-n}$. 2) For ...
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88 views

Evaluating difficult spectrum

Can anyone see how to show the spectrum of the bounded linear operator $T$ on $l^1$ defined by $$T((\alpha_j)) = (\alpha_j - 2\alpha_{j+1} + \alpha_{j+2})$$ is the cardioid $$\{(r, θ) : 0 ≤ θ < 2π, ...
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1answer
45 views

Condition for the unseparability of Banach Spaces

A basic question but I can't quite resolve it: Why is the following equivalent to unseparability of a Banach space X: For some uncountable set S $\subseteq$ X, there exists $\delta$ > 0 such that ...
3
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1answer
117 views

Completeness of $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle$

That's really embarrassing, however I need to ask it. I could not prove that the normed space $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle $ is complete (as a metric space), where $\| f\|_1 = ...
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0answers
105 views

Group action and Radon measure

Let $\mathscr M(\mathbb R)$ be the Banach space of complex-valued Radon measures on $\mathbb R$, and let $\pi$ be the action of $\mathbb R$ on $\mathscr M(\mathbb R)$. Let $\mathscr A$ denote a subset ...
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2answers
149 views

Using Neumann series to compute $T^{-1}$

Need help on how to show that $S$ satisfies the necessary condition for Neumann series. Here is what is given. $T\in B(X,X)$ where $X$ is a Banach space. Let $T: \mathbb R^3 \rightarrow \mathbb R^3$ ...
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1answer
82 views

Hilbert, Banach and isomorphism

I want to show that if linear mapping $L:B_1\rightarrow B_2$ is isomorphism of Banach space and $\|L(x)\|_{B_1} =\|x\|_{B_2} $ (surjective and isometry) so it consist that $L$ is isomorphism of ...
2
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1answer
123 views

Hamel basis dense in the unit sphere

I know that a Hamel basis can be dense in a Banach space (it was probably posted somewhere on this forum). I would like to construct a certain counter-example and doing this, I encountered the ...
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372 views

Completeness is property of the metric?

http://en.wikipedia.org/wiki/Banach_space From Wikipedia: In metric spaces, the completeness is a property of the metric. It is not a property of the topological space itself. If you move on to an ...
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0answers
79 views

(Real Analysis) Integration of two functions

Note ($\Omega,A,\mu)$ is a finite additive space. Let $f\in \bar S(A, \mathbb{R})$ and $y\in Y$ where $Y$ is Banach space. Prove $\int _\Omega yf d\mu = y \int _\Omega f d\mu$. Also, for any $X\in A$, ...
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551 views

How do I prove the completeness of $\ell^p$?

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ $\Rightarrow$...
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2answers
729 views

$C^\infty(R^n)$ is a Banach Space when equipped with topology of uniform convergence

Prove $C^\infty(\Bbb R^n)$ is a Banach Space when equipped with topology of uniform convergence. $C^\infty(\Bbb R^n)$ is space of all continuous functions that converge to $0$ at $\infty$. And, the ...
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1answer
48 views

Sequences in $\ell_1$

Consider $\ell_1$ as a dual of $c_0$. It is well known that there must exist sequences of elementes in $\ell_1$ which converge in the weak*-topology (wrt $c_0$) but not weakly. Can one give an example ...
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1answer
44 views

best constant for renorming a finite dimensional vector space

Consider an $n$-dimensional normed vector space $V$ with norm $p$. It is always possible to find a new norm $p'$ coming from an inner product and $p$ and $p'$ are comparable up to a factor $\sqrt{n}$. ...
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93 views

How to show that $(I - F)(U)$ is open, when $U$ is open and $F$ is a contraction?

Let $U$ be an open subset of a Banach space $E$ and let $F:U \to E$ be a contraction. Show that $(I - F)(U)$ is open. This is an exercise on page 9 of Fixed point theory and applications, Ravi ...
2
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1answer
314 views

Compact injections and equivalent seminorms

Let $V$ and $H$ be two Banach spaces with norm $\lVert \cdot \rVert$ and $\lvert \cdot \rvert$ respectively such that $V$ embeds compactly into $H$. Let $p$ be a seminorm on $V$ such that $p(u) + \...
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721 views

An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...
2
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1answer
61 views

Trying to show that equation has a single solution using Banach space Theorems

How do I show that $f(x) = \int_0^1 e^{-sx}\cos(\alpha f(s))~ds, $ $0\leq x\leq1$, $0\le\alpha\le1$ has a single solution. Using Banach space Theorems like Contraction mapping theorem? Thanks for ...
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124 views

Is $W_0^{1,p}(\Omega)\cap L^q(\Omega)$ Uniformly Convex?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $p<q$ with $p,q\in (1,\infty)$. Is $W_0^{1,p}(\Omega)\cap L^q(\Omega)$ Uniformly Convex with respect to the norm: $\|u\|=\|u\|_{1,p}+\|u\|_q$?...
2
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1answer
148 views

using uniform boundedness principle

I have a sequence of numbers $x_n$ that satisfy that for every $y_n \in c_0$ (when $c_0$ is a Banach space of all the complex sequences that satisfy $\lim_{n\rightarrow \infty }{a_n} =0$ ) the series ...
3
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1answer
841 views

Minimizing continuous, convex and coercive functions in non-reflexive Banach spaces

Let $X$ be a infinite dimensional real Banach space. If $X$ is reflexive, then any continuous, convex coervive function $f:X\rightarrow\mathbb{R}$ has a minimum value, that is assumed for some point $...
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Weak convergence in reflexive Banach space

Let $X$ be a reflexive Banach space. Let $T: X \to Y$ a linear operator. I want to show that: $$T \in \mathcal{L}(X, Y) \iff ((x_n \stackrel{w}{\rightharpoonup} x) \implies (T(x_n) \stackrel{w}{\...
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1answer
45 views

Maximun norm over the complex sequence

Is $C_0$ (the space of all the complex sequences that satisfy $\lim_{n\rightarrow \infty }{x_n} =0$ ) is a Banach space relative to the maximum norm ( $\|x\| =max|x_n| $) and pairwise operations ? ...
2
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1answer
310 views

Is it possible for a Strongly Convex function to be unbounded below?

Let $X$ be a non-reflexive Banach space and $f:X\rightarrow\mathbb{R}$ a $C^1$ function that is Strongly Convex, i.e. $$f(u)-f(v)\geq\langle f'(v),u-v\rangle+c\|u-v\|^2$$ where $c>0$ is constant. ...
3
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1answer
103 views

Is $W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$ complete?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain. Define $W=W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$. Is $W$ complete with respect to the norm $\|v\|=\|v\|_{1,p}+\|v\|_\infty$? If $u_n$ is a ...
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Is the result still true if we drop completeness? [duplicate]

I know how to prove the following exercise ( from Folland) : If $X$, $Y$ are Banach spaces. $T:X\rightarrow Y$ is a linear map such that $f\circ T\in\operatorname{dual}(X)$ whenever $f\in \...
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1answer
1k views

Isometric isomorphism

In the case that $L:B_1 \rightarrow B_2 $ is a linear mapping of Banach spaces and $L$ is a isometric isomorphism (bijection and $\|Lx\|_{B_1} = \|x\|_{B_2} $) can I say that $L\overline{L}= 1 $ is ...
4
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1answer
57 views

A certain element which makes functionals positive

Suppose $X$ is a possibly non-separable Banach space and let $X^*$ be its dual. Also, let $(f_n)_{n=1}^\infty$ be a sequence of [EDIT: linearly independent] norm-one functionals in $X^*$. Does there ...
0
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2answers
124 views

isomorphic properties

I need help with this proof: When $L$ is a isomorphic (bijection) linear mapping between two Banach spaces , in the case of both the spaces are Hilbert when using $L$, is it right to say that the ...
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2answers
410 views

From weak and weak star to norm convergence

I haven't found this yet and I'm somehow not sure if my idea is correct. The Problem: Let $X$ be a separable Banach-Space, let $x_k\to x$ weakly and such that for every $\lambda_k \to \lambda$ weakly-...
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1answer
169 views

Finite representability of $\ell_1$ and $c_0$

It follows from Krivine's theorem for a given Banach space $X$, either some $\ell_p$ or $c_0$ is finitely represented in $X$. Since finite dimensional $\ell_1$ and $\ell_\infty$-spaces are dual to ...
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1answer
140 views

Sequences in Banach spaces [duplicate]

I am very bad with proofs that ask you to show that "$\exists$ something..." because in most of them you have to explicitly show the something. The following question is one such. Any help will be ...
4
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1answer
567 views

Image of unit ball dense under continuous map between banach spaces

I am assuming that the following problem will require the open mapping theorem, or maybe the closed graph theorem. Any help that can be given will be deeply appreciated. The statement is the following:...
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710 views

Can a proper vector subspace of a Banach space be a countable intersection of dense open subsets?

Let $V$ be a complete normed space. Let $W$ be a proper vector subspace. Can $W$ be the intersection of a sequence of open dense subsets of $V$? If there exists an open dense proper vector subspace ...
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1answer
793 views

Dual space of Bochner space

Let $B$ be a refexive Banach space. I want to show that $$(L^2(0,T;B))^* = L^2(0,T;B^*)$$ and that the dual pairing is $$\langle F,f \rangle_{L^2(0,T;B^*), L^2(0,T;B)} = \int_0^T \langle F(t), f(t) \...
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210 views

Turning Banach space into Banach algebra

Given a Banach space, how can we determine if we can turn it into a Banach algebra or not?
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How to compute the norm of this particular bounded linear functional?

On the Hilbert space $l^2$, let $f$ be the functional defined by $$f(x):= \sum_{j=1}^\infty \alpha_j \xi_j$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $l^2$, where $a:= (\alpha_j)_{j=1}^\infty$ is a fixed ...
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281 views

A sequence converging weakly in $\ell^p$, for $p >1$ and failing to converge weakly for $p=1$

For $1 \le p < \infty$ and each index $n$, let $e_n \in \ell^p$ have $n$-th component 1 and all other componenets $0$. I want to show that $p>1 \Rightarrow \{e_n\} \to 0$ weakly in $\ell^p$ and ...
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0answers
61 views

A question about convergence in $L^p$. [duplicate]

Let $E$ be measurable and $1 \le p \le \infty$. Suppose $\{f_n\}_{n \in \mathbb{N}}$ all measurable and $\{f_n\}_{n \in \mathbb{N}} \to f$ pointwise a.e. $E$. For $p$ as above, I want to show that: $$...
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1answer
105 views

Hypervolume of a $N$-dimensional ball in $p$-norm

Suppose I have a N-dimensional ball with radius R in p-norm: $$ \sum_{n=1}^N |x_n|^p = R^p $$ Is there a closed formula for its (hyper)volume? I can't find anything. If there isn't, can we at least ...
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1answer
118 views

Clarification of Reed and Simon proof of the open mapping theorem

I was reading Reed and Simon Methods of Mathematical Physics Volume 1 and have a question about a small their proof of the open mapping theorem for Banach spaces. Let $T:X\rightarrow Y$ be a bounded ...