Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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A subspace isomorphic to $C[0,\alpha^{\omega}]$.

Let $\omega\leq \alpha<\omega_1$ ordinal, $Y$ Banach space, then $\ell_{\infty}^*\otimes_{\pi}Y$ has a subspace isomorphic to $C([0,\alpha^{\omega}])$? For me it would be nice if the answer was no. ...
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41 views

A proposition in C*-algebra

Problem: Let $A$ and $B$ be C*-algebra and $\varphi:A \rightarrow B$ be a contractive completely positive map. $A_{\varphi}=\{a\in A: \varphi(a^{\ast}a)=\varphi(a)^{\ast}\varphi(a)$ and ...
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44 views

invariant subspace partition

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf on page 327 Rudin says that M and M' are invariant subspaces. I'm guessing he means non-trivial so how does ...
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46 views

Bochner measurability

I have the following problem. Let $(\Sigma, \Omega, \mu)$ be a measure space and let $X$ be a Banach space. Take a function $f \colon \Omega \rightarrow \mathbf{B}(X)$, which takes values in space of ...
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1answer
40 views

Prove a functional to be differentiable

Consider the functional $I:C[a,b]\rightarrow\mathbb{R}$ given by $$I(x)=\int_{b}^{a}x(t)dt.$$ Prove that $I$ is differentiable and find its derivative at $x_0\in C[a,b]$. The answer just says that it ...
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1answer
112 views

Why the tempered distribution is zero?

My question is derived from the proof of the equation $\Delta f=f$ which has no nonzero solution in $\mathscr{S}'(\mathbb{R}^n)$. The ideal to solve this equation is to use the Fourier transform. By ...
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1answer
44 views

Bessel Sequence proof check.

I have a similar question to definition of Bessel sequence, where it was solved using Banach-Steinhaus. A sequence $\{f_k\}_{k=1}^{\infty}$ is called a Bessel sequence in a Hilbert space ...
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1answer
97 views

Dual space of $l^1$

I m taking a course in functional analysis. The book state that the dual space of $l^1$, the set of real valued absolutely summable sequence, is $l^\infty$. Can anyone explain why the dual space of ...
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42 views

How do they call the topological tensor product that classifies operators from Hilbert space?

Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the ...
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66 views

Sequence of connected sets converging to disconnected set

Let us have the disconnected set $\mathcal{X} = \{0\} \cup [\underline{x},\overline{x}]$ and its approximation $\mathcal{X}_{\nu} \subseteq [0,\overline{x}]$, with $0 > \underline{x} > ...
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1answer
50 views

Rudin's proof of invariant subspace existence

I have questions about Rudin's proof of invariant subspace existence. On page 327, point 12.27, How does he get that $Tx=TE(\omega)x$, and How does he know $E(\omega)$ is not the zero map? ...
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1answer
74 views

Integral convergence involving Lp and Sobolev spaces

Quick question, any contribution or hint would be appreciated: How does it follow that: $$\lim\limits_{k \rightarrow \infty}\int_{\Omega}a(u_{k})\frac{\partial u_{k}}{\partial x}\frac{\partial ...
4
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2answers
201 views

Subspaces in the image of compact operator

Let $X$ and $Y$ be some infinite dimensional Banach spaces. Let $T:X\longrightarrow Y$ be some compact linear operator. It is easy to understand that $T$ cannot be surjective: the Open Mapping Theorem ...
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1answer
120 views

Rudin functional analysis problem

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf 1) page 326, he says that if ST=TS, then S commutes with f(T). He has previously shows that if S commutes ...
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36 views

Why Steiner Symmetrization makes a measurable set to a measurable one?

I find the Steiner Symmetrization is very useful in proving that the Hausdorff measure coincide with Lebesgue in the Euclidean space. However, I never saw anybody mention that the Steiner ...
2
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1answer
54 views

Pointwise boundedness

I know if a real valued function $g$ is continuous on a closed and bounded interval $[a,b]$, then it is bounded. However, I am not sure whether the following holds for sequences of functions: ...
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1answer
58 views

Are $C^0[a,b]$ and $C^0[0,1]$ isometrically isomorphic?

Consider $C^0[a,b]$ and $C^0[0,1]$, each equipped with the $L^1$-Norm. Are these (out of curiosity) isometrically isomorphic?
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2answers
115 views

How to show that the monomials are not a Schauder basis for $C[0,1]$

why the monomials are not a Schauder basis for $C[0,1]$? $p_n(x)=x^n$ such that $(p_n)$ does not form a Schauder basis for $C[0,1]$ span$\lbrace p_n : n\ge 0\rbrace$ is dense in $C[0,1]$ by ...
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1answer
79 views

The excision theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". I can not understand the proof of the excision theorem in the fundamental facts of the book. Theorem 1.4.10(Excision) Let ...
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2answers
109 views

Normal Self-Invertible Operator is Self-Adjoint

If $T\in B(H)$ for some Hilbert space $H$, is a normal operator and $T^2=I$, then $T=T^*$. It seemed simple when I first saw the claim, but I'm having trouble showing it. I know that it implies ...
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1answer
75 views

An exercise in C*-algebra

Let $A$ be a C*-algebra, $\phi$ be a pure state and $L=\{a\in A:\phi(a^{\ast}a)=0\}$, how to prove that $L+L^*\subseteq ker\phi$. ($L^*=\{a^{\ast}: a\in L\}$) I think it is an easy exercise, ...
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23 views

Morphism of $G$-modules, representation in banach space

We have some group $G$ and it representation in banach space $X$. It means that there is a group-homomorphism $T:G\longrightarrow \mathrm{GL}(X)\cap \mathfrak{B}(X)$, where $\mathfrak{B}(X)$ is a set ...
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1answer
42 views

A question about $C^\ast$-algebra

In Kadinson's book Fundamentals of The Theory of Operator Algebra, when the author proved the Theorem 7.2.1, he let $V$ be an extreme point of the unit ball of $C^\ast$-algebra $\cal{U}$, $h$ be a ...
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88 views

Convergence of sequences of sets

Assume we have the indicator function $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and the sequence of its approximation functions $\{ I_{\nu}(x) ...
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1answer
58 views

Is there example for isomorphic closed subspaces of a Banach space with non isomorphic quotient?

$Y_1$ and $Y_2$ are closed subspaces of a Banach space X and $Y_1 \simeq Y_2$. I can't find a way to show $X/Y_1 \simeq X/Y_2$ and it made be think that it's not true. Is there a counter example?
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1answer
129 views

Why are so many inequalities and estimate in PDEs?

I want to study PDEs, but when I read some PDEs books, I notice that there are so many inequalities here and always do estimates. I really want to know why there are so many inequalities here and ...
2
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2answers
612 views

Weak convergence in a Hilbert Space

What does it mean for a sequence $\{f_n\}_{n=1}^\infty\subseteq H$ to converge weakly? I know it means that it converges in the weak topology and I've read a few definitions of weak topology which all ...
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1answer
97 views

Show the Banach-Mazur distance is reached for finite-dimensional Banach spaces

Let $X,Y$ be isomorphic Banach spaces. The Banach-Mazur distance: $d(X,Y)=\inf \{\| T\| \| T^{-1}\| : T:X\rightarrow Y \text{is an isomorphism} \}$ can be rewritten as: $d(X,Y)=\inf \{\| T^{-1}\| ...
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68 views

Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
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36 views

operator onto-theorem

I have this theorem: Let $V$ a Banach space, reflexive,separable, and let $A$ an operator monotonic, bounded, semi-continuos, coercive. Then, $A$ is onto. Where we can find the proof of this ...
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51 views

Bounded inverse theorem.

http://planetmath.org/boundedinversetheorem referring to this proof i don't get the final statement: "$T^{-1}$ is continuous, i.e. bounded". I know that the boundness would be surely true if $T^{-1}$ ...
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268 views

Does $f(x)\in L^1$ imply that $f'(x) \in L^1$?

Let $f(x)$ be defined for all real numbers differentiable function of one variable.We know that: $$\int_{-\infty }^{+\infty } |f(x)| \, dx\neq +\infty$$ Problem is to resolve if it is possible or not ...
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1answer
73 views

When are two Banach spaces equal?

Suppose $X$ and $Y$ are two Banach spaces. If there is an isometric isomorphism, then we can say that these two spaces are same. Is there any other condition which will say that these two spaces are ...
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1answer
27 views

References about Nemytskii Mappings

I need some references about Nemytskii Mappings. Can anyone tell me some textbook about it? I am reading chapter 2 of this text www.math.tifr.res.in/~publ/ln/tifr81.pdf . And I need more results ...
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73 views

Reference request for nonlinear functional analysis notes.

I'm currently trying to read a paper on Fixed Points of Asymptotic Contractions" by W.A. Kirk. A small excerpt can be seen here. Those with accounts on the Elsevier page can see the whole content. ...
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64 views

Norm of multiplication operator

I have that $(X,\Omega,\mu)$ is a sigma finite space, and I have that $g$ is a measurable function. Assume that $fg\in L^p$ for all $1\leq p\leq \infty$. I want to show that $g\in L^\infty$. My idea ...
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163 views

$l^\infty(I)$ and $l^\infty(J)$ isometrically isomorphic with $|I| \not= |J|.$

Is it possible for $l^\infty (I)$ and $l^{\infty} (J)$ to be isometrically isomorphic with the cardinality of $I$ not equal to the cardinality of $J$? I'm able to show that if $1\le p < \infty,$ ...
2
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2answers
307 views

Gaussian function as a mollifier?

I am wondering if the function $f_{n}(x)=\sqrt{\frac{1}{2\pi n^2}} exp(\frac{-x^2}{2n^2})$ can be viewed as a mollifier. I would like to prove that if $g$ is a continuous and bounded function then ...
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2answers
44 views

Convolution product

How can we prove that if $f$ is compactly supported and $g$ is periodic with period $P$ then $f*g$ exists and is also P-periodic ? thanks.
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1answer
106 views

An equality that holds with $v_t \in L^2(0,T;L^2(\Omega))$ but its proof requires $v_t \in L^2(0,T;H^1(\Omega))$

Let $Q=(0,T)\times \Omega$. For all $\varphi \in C_c^\infty(Q)$ such that $0 \leq \varphi \leq 1$, the following holds $$\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla v|^2 \varphi \varphi_t - ...
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1answer
124 views

Question about perpendicular complements in Banach spaces

Let $V,W$ be Banach spaces with $T : V \to W$ a bounded linear transformation. Let $T^* : W^* \to V^*$ be the standard adjoint map on the dual spaces. That is, for $g \in W^*$, we have $[T^*g](v) = ...
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1answer
42 views

Cardinality of maximal subsets with some property

Let $X$ be a normed space. Is it true that all maximal (with respect to "$\subset$") subsets $D\subset X$ with the following property: $$ \|x-y\| \geq1 \textrm{ for } x\neq y, x,y\in D, $$ are of the ...
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1answer
65 views

How to explain a theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". In the fundamental facts, it introduce the Noncommunicative Lusin's theorem: Let $A\in B(H)$ be a nondegenerate C*-algebra ...
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2answers
254 views

About the range of an operator and its adjoint

I need a hand with the proof of this result: If we have an operator between Banach spaces $T:X\to Y$, with closed range, then the adjoint operator $T^*:Y^*\to X^*$ has also closed range. Thanks in ...
3
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3answers
110 views

Pointwise Convergence in $L^1$ norm

Suppose we have a sequence of functions $\left\{f_n\right\}_{n=1}^\infty\subseteq C^1([0,1])$ and $f_n\to f\in C([0,1])$ in the $L^1$ norm and $f_n'\to g\in C([0,1])$ in $L^1$. Does it follow that ...
5
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1answer
92 views

Spectral theorem in Quantum Mechanics

In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
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2answers
110 views

Is the sequence $(x_n)$ convergent in the space $L_1(0,1)$

Is the sequence $(x_n)$ convergent in the space $L^1(0,1)$ ? $x_n(t)= n^2 t^n (1-t^2)$ for $n\in\mathbb{N}$. norm: $\|x\|=\int_{(0,1)} \left|x(t)\right| \; dt$ I think it should ...
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54 views

$2$-Normed Spaces

Someone suggested today that $2$-normed spaces are actually equivalent to normed spaces. Can anyone who's familiar with the topic provide a counterexample? (I can't access Gähler's original paper ...
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2answers
28 views

Extending Continuous Basis

It is given $(k-d)$ continuous vector-valued functions $K_1,\dots,K_{k-d}:\mathbb{R}\mapsto\mathbb{R}^k$, with $d\leq k$. Suppose that for all $x\in\mathbb{R}^k$, the set ${\cal ...
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26 views

$T$ is injective iff $T(H)$ linearly independent.

Let $T:X\to Y$ be a linear transformation and $H$ be a hamel basis of $X$, then prove that, $T$ is injective iff $T(H)$ linearly independent.