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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Bounded Measurable Functions: Pointwise Limit vs. Uniform Limit

Agreement All notions are up to null sets. Limits are meant by simple functions. Problem Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. Consider bounded measurable ...
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Generalized Riemann Integral: Uniform Convergence

Disclaimer This thread is meant to record. See: Answer own Question And it is written as question. Have fun! :) Reference This thread is related to: Generalized Riemann Integral: Nonexample? ...
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Banach Space E is the direct sum of two closed subspace L and M, M is finite dimensional, T is a bounded linear operator from E to E, please prove …

Banach Space E is the direct sum of two closed subspace L and M, M is finite dimensional, T is a bounded linear operator from E to E, please prove that T(E) is closed subspace of E if and only if T(L) ...
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$L^p$ is a quasi normed space for $0<p<1$ [duplicate]

I know that $L^p$ is a vector space for $p>0$ and a normed space for $p \geqslant 1$ now I need show that for $ 0<p<1$ and $f,g \in L^p$ exist $K \in \mathbb{R}$ such that $||f+g||_p ...
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23 views

Banach Spaces: Improper Riemann Integral

Disclaimer This thread is related to: Stone's Theorem Definition Given a measure space $\Omega$ and a Banach space $E$. Consider functions $F:\Omega\to E$. Denote the measurable subsets of finite ...
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1answer
51 views

Recapitulated: Stone's Theorem Integral

This problem grew out from: Stone's Theorem Integral For a definition, a nonexample and a comparison see: Generalized Riemann Integral: Definition Generalized Riemann Integral: Nonexample ...
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Finding a norm making a subspace dense

Suppose $V$ is a (real or complex) vector space and $W$ is a subspace of $V$. Under what conditions is there a norm on $V$ making $W$ a dense subspace of $V$? That $V$ and $W$ have the same ...
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47 views

Adjoint of completely continuous operator is completely continuous

In the proof of the fact that the adjoint operator $A^\ast$ of a completely continuous linear operator $A:E\to E$ mapping a Banach space into itself is also completely continuous on $E^\ast$ endowed ...
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When is the spectra of a function composed completely of eigenvalues?

For a linear operator $T$, define its spectrum to be $\sigma(T)$. Also define $L = \{\lambda \mid Tv = \lambda v$ for some $v\in B$. Certainly, $L \subset \sigma(T)$, and if $T$ is finite ...
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Eigenvalues of Sub-Matrix Formed from subset of Columns

I have an n-by-p matrix $X$ and I consider the eigenvalues of the p-by-p matrix $X^{'}X$. Let's denote the largest and smallest eigenvalues of $X^{'}X$ with the usual notation $\lambda_{1}(X^{'}X)$ ...
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37 views

Can you show this space isn't complete?

Let $X=C^0([0,1])$ and $||\cdot||:X\to\Bbb R$ be defined as $$||f||=\max_{x\in[0,1]}x^2|f(x)|.$$ Show that $||\cdot||$ isn't a Banach space. (I can't find any Cauchy sequence that does not converge. ...
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38 views

Show the space X is incomplete

A problem comes from the "Optimization by vector space methods". Luenberger p.34 Let $X$ be 1. the space of continuous functions on [0,1] 2. its norm is defined by $||x|| = \int^1_0|x(t)| dt$ So, 1. ...
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16 views

Projections onto disjoint spectra in functional calculus

Theorem 6 part (i) of Lax's Functional Analysis book (Chpt 17) states (paraphrased) Suppose that the spectrum of $M$ can be decomposed as the union of $n$ pairwise disjoint closed components: ...
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26 views

A question in functional analysis about bounded linear operator.

Suppose $Banach$ Space $E$ is the direct sum of its closed subspaces $L$、$M$, and $M$ is finite-dimensional, $T$ is a bounded linear operator from $E$ to itself. Prove that $T(E)$ is a closed subspace ...
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1answer
29 views

Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims $$u = strong - \lim_{\epsilon\to 0} ...
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33 views

If $X\subset Y$ then $X^*\subset Y^*$

Is the following true, If $X$ and $Y$ are Banach spaces and $X\subset Y$, then $X^*\subset Y^*$. One argument for this is the following let $i:X\to Y$ be the identity map which implies its one to ...
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43 views

Power Series: Derivative

Given a Banach space $E$. Consider a series: $$|t|\leq R:\quad\sum_{k=0}^\infty A_k t^k\quad(A_k\in E)$$ Is there an elegant proof of: $$\left(\sum_{k=0}^\infty A_k t^k\right)'=\sum_{k=0}^\infty A_k ...
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C*-Algebras: Contractive Morphism

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with $\mathbb{1}_\mathcal{A}\in\mathcal{A}$. Consider an algebraic morphism $\pi:\mathcal{D}\subseteq\mathcal{A}\to\mathcal{B}$ with ...
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149 views

Prove that the normed vector space $(S_F,\|\cdot\|_1)$ is not Banach.

$S_F$ is the space of real sequences $\mathbf a=(a_n)_{n=1}^{\infty}$ such that every sequence $\mathbf a\in S_F$ is eventually zero. $\|\cdot\|_1$ is the norm defined as $\|\mathbf ...
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12 views

The example of a mapping is homomorphic but not isomorphic

The example of two Banach space A,B such that the mapping T:A→B, is onto , one-to-one and homomorphic but not isomorphic i.e ∥T(x)∥≠∥x∥. I think there are two norm spaces, such that norm does not ...
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25 views

Non-Reflexive Spaces

I read the following example of non-reflexive spaces which I do not understand. Let $X:=C([0, 1])$ be the space of continuous function on $[0, 1]$. It is mentioned that the dual of this space, $X^*$, ...
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Example (?) of a Banach space containing an uncomplemented copy of itself

I was wondering the following: Background Question: Does there exist a Banach space $X$ which contains a copy $X_0 \subset X$ of itself that is not complemented? By "$X_0$ is a copy of $X$", I ...
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Sobolev spaces and Cauchy sequences with respect to $L^2$-norm.

Let $z^n=(u^n,w^n,\phi^n)$ be a sequence in $H=H_*^1(0,\ell)\times H_0^1(0,\ell)\times H_*^1(0,\ell)$, where $H^1(0,\ell)$ and $H_0^1(0,\ell)$ are the usual Sobolev spaces and $H_*^1(0,\ell)=\{f\in ...
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32 views

Surjective Operators

Let $X$ be a separable Banach space. Show that there is a surjective linear operator $P: L^1(\mathbb N) \to X$ and a subspace $S \subset L^1(\mathbb N)$ such that $$\|P(x)\| = \inf_{y \in S} ...
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29 views

How to see this is an isometry

Let $X$ be a separable Banach space, and $(f_n)$ be a countable dense subset. Recall that for each $f_n$ there exists a linear functional $l_n \in X^*$ such that $\|l_n\| = 1$ and ...
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1answer
38 views

examples on double dual spaces

I am looking for examples on double dual spaces as I know $\ell_p $ is a double dual of $\ell_p$ for $1<p,q<\infty$. $\mathcal L_p $ is a double dual for $\mathcal L_p$ for ...
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How to find linear operators

Hello can you help me on how to find the linear operator of an identity function of a mormed linear space of all polynomials in a unit interval. $T: X \rightarrow X$ by $(Tx)(t)= x^1(t) t\in[0,1]$ ...
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26 views

How to show this space is NOT reflexive

Consider the Banach space $X$ of null sequence whose elements are complex sequence which converges to $0$. In addition the norm is defined as $$\|(a_1, \dots, a_n)\| := \sup_n |a_n|.$$ Show this ...
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49 views

If $E$ is not complemented in $X$, is $E \oplus \{0\}$ not complemented in $X \oplus Y$?

Question: Let $X$ be a Banach space, and let $E \subset X$ be a closed subspace such that $E$ is not complemented in $X$. Does it follow that $E \oplus \{0\}$ is not complemented in $X \oplus Y$, ...
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21 views

Proof of “Dual normed vector space is complete”

http://en.wikipedia.org/wiki/Dual_norm As in the introduction of dual norm by Wiki, it says dual normed space $X'$ is always complete. How to prove that? or at least explain that? We all know the ...
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28 views

Weak-* Convergence of Linear Functionals

Let $X$ be a Banach space and $f_n$ be a sequence in the dual space $X^*$ such that for all $x \in X$, the sequence $f_n(x)$ converges. Show that $(f_n)$ exists a weak-* limit $f \in X^*$. In ...
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Literature: Derivations in C*-Algebras

Do you have some nice reference for dynamical systems in C*-algebras (including discussion of their derivations!) like notes, papers, books, etc.?
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The dual of the Banach space $C(\Omega)$

It is well-known that the dual of the Banach space $C([0,1])$, i.e. the space of all continuous functions on the interval, is the space of all functions of bounded variation on the interval, ...
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Completing the solution, lipschitz maps inducing other maps

Let $(X, d)$ be a metric space, $(E, || \cdot ||)$ a Banach space, $(AE(X), || \cdot ||)$ - as described below. I've already proven that for any Lipschitz function $u: X \rightarrow E $ there exists ...
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$(\lim x_n)y = \lim (x_n y)$ when $x_n, y \in B(H)$

Please give me a hint to prove $(\lim x_n)y = \lim (x_n y)$ when $x_n, y \in B(H)$ for every n. Thanks in advance.
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Banach spaces with a bounded linear functional constant on some normalized Hamel basis

Let $V$ be a normed vector space over $\mathbb R$. For a normalized Hamel basis $\mathcal B$ of $V$, consider the linear functional $f_{\mathcal B}:V\to\mathbb R$ taking constant value $1$ on ...
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Banach space is sum of $ker f$ and $X \ ker(f)$.

I'm trying to show that if $f$ is an element of the dual space $X^*$ of a Banach space, $X$, and $x_0 \in X-ker(f)$, then every element in $X$ can be expressed as $x = \lambda x_0 + y$ with $y \in ...
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Linear Continous Mappings

I am trying to prove the following Theorem: where $E$ and $F$ are normed vector spaces over the field $\mathbb{R}$ equiped with topologies introduced by means of their norms. I am not able to ...
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Finitely representable space

Background: Let $E,F$ be Banach spaces with $F$ be finitely representable in $E$, and separable. That is, for all finite dimensional subspaces $M\subset F$ there is a $(1 + \epsilon)$-isomorphism ...
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26 views

Showing that the dual of Banach space $l^1$ is $l^{\infty}$

I'm trying to show that the dual of Banach space $l^1$ is isometrically isomorphic $l^{\infty}$. I've defined a linear map $F: (l^1)^* \to l^{\infty}$ by $F(y)(x) = \sum x_n y_n$. So far I've shown ...
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How does a Function solving a Functional Equation changes with respect to a change of a Parameter of that Equation?

I want to see how a function solving a functional equation changes with respect to a change a parameter of the functional equation. In particular, let $C(X)$ be a Banach space with continuous and ...
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right inverse and supplement of kernel in a banach

For $T \in L(E,F)$ continuous surjective linear operator between Banach spaces $E$ and $F$ we have that : $Ker(T) $ admits a closed complement $L$ in $E \implies T$ admits a continuous right ...
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143 views

Spectrum of a nilpotent operator

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator such that $A^n=0$ for some $n\in \mathbb{N}$. Is the spectrum of $A$ finite, countable ?
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Existence Adjoint Operator: Boundedness?

Context This would make the check on the GNS construction much more simple. Problem Given a Hilbert space $\mathcal{H}$. Consider a merely linear operator $A:\mathcal{H}\to\mathcal{H}$. Suppose ...
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95 views

Generalized Riemann Integral: Nonexample?

Reference For a comparison of integrals see: Uniform Integral vs. Riemann Integral Definition Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. (In fact, a Hausdorff TVS ...
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Are the invertible elements of a Banach algebra closed in the set of left-invertible elements?

Let $A$ be a unital Banach algebra. Denote by $\mathrm{Inv}(A)$ the invertible elements in $A$, and $\mathrm{Inv}_\ell(A)$ the left-invertible elements. That is, $a \in \mathrm{Inv}_\ell(A)$ if and ...
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Understanding dual spaces

My professor asked me to give him an example about dual spaces from our real life so I was thinking if we take the set of all our actions as the space X can the space of all our thoughts be the dual ...
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Proof Riesz Representation Theorem (bounded linear functional in Lp)

I have a little problem with this proof (I'm using Royden), can you help me? Let $F$ be a bounded linear functional on $L^p$, $1 \leqslant p \leqslant \infty$. Then there is a function $ge \in L^q$ ...
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35 views

Weak Convergence and Weak Topology

In discussing weak topology of a normed space $X$, a lemma is given as follows. If $(x_n)$ is a sequence in $X$ converging weakly to $x$, then $x_n$ is bounded. I understand the proof of this ...
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Continuous Strong-Strong Implies Continuous Weak-Weak

Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak. I can see that $T$ ...