Tagged Questions

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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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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1answer
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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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1answer
33 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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1answer
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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1answer
25 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
27 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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1answer
32 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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39 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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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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0answers
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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23 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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19 views

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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1answer
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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1answer
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
32 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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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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1answer
47 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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1answer
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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1answer
25 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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1answer
16 views

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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1answer
31 views

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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17 views

$(\lim x_n)y = \lim (x_n y)$ 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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About a family of (possibly uninteresting) Banach spaces

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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2answers
18 views

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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1answer
18 views

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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1answer
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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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1answer
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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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142 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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Generalized Riemann Integral: Nonexample?

Definition Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. (In fact, a Hausdorff TVS should suffice.) Consider functions $F:\Omega\to E$. Define the generalized Riemann ...
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Boundedness on bounded sets [closed]

Let $\Omega\in\mathbb{R}^n$, $u\in L^\infty(I;H^1_0(\Omega))$, where $I$ is an interval, and $g:H_0^1(\Omega)\mapsto H^{-1}(\Omega)$ bounded on bounded sets. How to prove that with this assumption we ...
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1answer
51 views

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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1answer
29 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 $T$ ...
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0answers
26 views

Dual of a locally convex space

Let $X$ be a normed space. Suppose $E$ is a subset of $ X^*$ (The space of continuous linear functionals). For every $\phi\in E$, define seminorm $p_\phi: X\to [0,\infty)$ such that $p_\infty (x)= ...
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1answer
59 views

Banach Spaces: Uniform Integral vs. Riemann Integral

Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. Consider functions $F:\Omega\to E$. Predefine a simple integral: ...
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1answer
43 views

If $f(x)$ is close to $0$ then necessarily $x$ is close to $\ker f$?

Suppose that $X$ is a real Banach space and $f:X \to \mathbb{R}$ is a continuous linear functional. Is it true that for any $\varepsilon>0$ there is a $\delta>0$ such that for any $x \in X$ we ...
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1answer
50 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
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1answer
29 views

Inner products of weakly convergent sequences

I have a weakly convergent sequence in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$. I want to show that there is a sequence $v_k\rightharpoonup v$, such ...
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2answers
83 views

Best approximation for a closed set in a finite dimensional normed space

First of all I'd like to mention that it is a part of my home work so I'd like if you won't give the answer itself, but try to guide me into it. I've been losing my mind for the last couple of hours ...