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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30 views

Prove of some properties about unitary operators [closed]

Let $X$ be a hilbert space and $T\in L(X)$ be an unitary operator. Show (1) $\sigma(T)\subset\{\lambda \in \mathbb C:|\lambda|=1\}$ (2) for $\lambda \in \mathbb C$ with $|\lambda|\neq1$ holds: ...
3
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31 views

Using Mac Shane's Lemma

Let $I \subset \mathbb{R}^{N}$ be a convex, bounded open set with Lipschitz boundary $\partial I$. Let $\lbrace u_{n} \rbrace_{n}$ and $u$ be such that $$ u_{n} \rightharpoonup^{*} u~~ \text{ in }~ ...
6
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59 views

Nuclear spaces vs Banach spaces

The Wikipedia article on nuclear spaces say the following: "There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: ...
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17 views

Showing map is isometry between Banach quotient space

I have a closed subspace $Y$ of a Banach space $X$ and a map $T: X'/Y^{\circ} \to Y'$ given by $[f] \to f|_y$. The norm in $X'/Y^\circ$ is given by $\|[f]\| = \inf \{ \|f-h\| : h \in Y^\circ \}$. I'm ...
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35 views

What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an ...
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2answers
36 views

Extending linear functional in non-unique way

I'm trying to find an example of when the extension of a functional in the Hahn-Banach theorem is not necessarily unique. I'm looking at the space of continuous functions on $[0,1]$ and I'm trying to ...
2
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1answer
39 views

Finite rank volterra operator

I am wondering when a Volterra integral operator $V_K:L_2(0,1)\to L_2(0,1)$ is a finite rank operator: $$V_Kf=\int_0^xK(x,y)f(y)dy$$ thanks in advance for your help
5
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1answer
60 views

Showing $\int f(x)e^{nx}$ implies $f(x) = 0$

I'm trying to show that if $f$ is a continuous function on $[0,1]$ and $\int_0^{1} f(x)e^{nx}\,dx = 0$ for all $n = 0,\ 1, 2,\ \dots$, then $f(x) = 0$. I'd like to use the Weierstrass approximation ...
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1answer
32 views

Continuous operator between Banach spaces, closed range

I have some problems proving the following: $T: X \rightarrow Y$ is a continuous, linear operator between Banach spaces. Prove that $T$ is surjective $\iff$ $T^* : Y^* \rightarrow X^*$ is injective ...
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1answer
23 views

Is $Tf=x_0+\int_0^tf(t)(1-f)\,dt$ a Lipschitz function?

Consider the Banach space $C[0,1]$ with the uniform norm and the operator given by $(T(f))x=x_0+\int_0^xf(t)(1-f(t))\,dt$ for $x\in[0,1]$ and $x_0\geq0$. I want to show that there is a number ...
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32 views

Use Poisson summation formula to prove Gaussian sum formula

The Poisson summation formula states that for any Schwartz function $f$, $\sum\limits_{k\in\mathbb{Z}}f(k)=\sum\limits_{k\in\mathbb{Z}}\hat{f}(k)$, where $\hat{f}$ is the Fourier transform of $f$. The ...
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0answers
23 views

Continuity of the dual product reloaded

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $(x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R}$ is strongly$\times$strongly continuous in $X\times X^*$. That does not ...
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1answer
21 views

Can the equivalence of isometry and unitarity of a linear operator be extended to infinite dimensions?

I was wondering whether it is possible to extend the following standard theorem to infinite dimensional Hilbert spaces? Let $M\in \mathbb{C}^{n\times n}$ arbitrary. The matrix $M$ is unitary if and ...
3
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34 views

Conway base 13 function, Darboux functions and unbounded linear functionals

After reading many interesting posts here about Darboux functions and Conway's base 13 function (http://en.wikipedia.org/wiki/Conway_base_13_function) I have some questions that I don't seem to answer ...
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1answer
10 views

Proving that $(Sf)(x)=1+\int_0^x t^2f(t)\,dt$ is $K$-Lipschitz

I want to show that $S:C[0,1]\to C[0,1]$ where $$(Sf)(x) = 1+\int_{0}^{x} t^2\cdot f(t) \,dt$$ is a Lipschitz map for $x\in[0,1]$. There I use the Euclidean norm on $\mathbb{R}$ and the uniform norm ...
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0answers
24 views

Invariant subspace of bounded self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
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1answer
31 views

Invariant subspace of self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
2
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1answer
28 views

Existence solution pendulum equation using the contraction principle

How can I prove that the differential equation $\ddot{x}(t)=-(g/\ell)\sin (x(t))$ must have a solution by using the contraction principle (by Banach). The numbers $g$ and $\ell$ are fixed constants ...
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1answer
40 views

An exercise showing that $l^1$ is not the dual of $l^\infty$

there is a well known fact that $l^1$ is not the dual of $l^\infty$. An exercise Folland's Real analysis serves as an example for this.(Page 192 ex 19) Define $\phi_n \in (l^\infty)^*$ by ...
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45 views

When $1 \le p \le \infty, p\ne 2$, $L^p$ space is not a Hilbert space

It suffices to show that when $1 \le p \le \infty, p\ne 2$, $L^p$ norm does not arise from an inner product.(there is a hint saying that we can use the parallelogram law) I can proof a special case ...
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1answer
44 views

Is there a $f\in C[0,1]$ such that $f(x)=\frac12 \sin (f(1-x))$

Is there a $f\in (C[0,1],\lVert\cdot\rVert_{\infty})$ such that $f(x)=\frac12 \sin (f(1-x))$? I feel like you need to apply Banach´s fixed point theorem, which means it suffices to prove that ...
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1answer
33 views

Special integrands in the calculus of variations

Most techniques in the calculus of variations that I know of, deal with integrands of the form $W(x, \phi(x), \nabla \phi(x)): \Omega \times \mathbb{R}^n \times \mathbb{R}^{n \times n} \to ...
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1answer
23 views

Density of continuosly differentiable function in space of continuous functions

Let $C([0,1])$ be the set of all real continuous functions with the standard supremum norm. Let $C^1([0,1])$ be the set of all real continuosly differentiable functions on $(0,1)$ such that the ...
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0answers
7 views

Approximating the weak gradient of the constant function in $L^p$

I want to find a sequence $u_n:(0,\infty) \to \mathbb{R}$ such that $u_n \to k$ pointwise and $\nabla u_n \to 0$ in $L^p$, where $k$ is the constant function equal to $k \in \mathbb{R}.$ Will the ...
2
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2answers
43 views

Is $T:C[0,1]\to C[0,1]: Tf(x)=\frac12 x f(x^2)+1$ a contraction?

I am a non-mathematician (a physicist) who wants to find a continuous function on $[0,1]$ that satisfies the relation $f(x)=\frac12 x f(x^2)+1$ with $0\le x\le 1$. I need it for one of my models. ...
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1answer
41 views

Completion of a Banach space with respect to a different norm

Let $(X,|\cdot|_X)$ be a Banach space. Define a space $Y$ as the completion of $X$ under a norm $$|u|_Y = |u|_X + |Tu|_Z$$ where $T:X \to Z$ is a linear continuous map where $X \subset Z$ is a ...
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1answer
26 views

Examples of locally convex space

In "A course in Functinoal analysis" by conway, I want to show that Let $X$ be a completely regular and $C(X) = $ all continuous function from $X$ into $\mathbb F$. If K is compact subset of $X$, ...
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20 views

Does the convolution property suffice to show $\hat \chi*\hat \chi=\hat \chi$?

Given a compact and sufficiently regular set $\Omega\subset\mathbb{R^n}$ and its characteristic function $\chi=\chi_\Omega$, I would like to conclude that the (inverse) Fourier transform ...
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0answers
14 views

Eigenelements of Neumann Laplacian satisfy $\sum_{k=1}^\infty |(u,\varphi_k)_{L^2}|^2 \lambda_k^{-\frac 12} < \infty?$

Let the eigenvalues of Neumann Laplacian on a bounded open domain be given by $0 = \lambda_0 \leq \lambda_1 \leq \lambda_2 ...$ associated to eigenfunctions $\varphi_0, \varphi_1, ...$. Let $u \in ...
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21 views

Detail in a proof about energy minimizing harmonic maps

Let $u\in H^1(B_1;S^k)$, where $$B_1:= \{x\in\mathbb{R}^n: \lvert x\rvert<1\}\\ S^k:=\{x\in \mathbb{R}^{k+1}: \lvert x\rvert=1\}. $$ Suppose $u$ is a minimizer for the Dirichlet energy functional ...
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1answer
39 views

Prove that if ,$||f||^2 = A\sum_{j}|<f, \phi_j>|^2 $ then $f = \sum_{j}<f, \phi_j> \phi_j$

Let $\phi_k$ be some sequence of real functions in an infinite Hilbert space $H$ such that there exists $A \in \mathbb{R}$ such that for all $f \in H$ ,$||f||^2 = A\sum_{j}|<f, \phi_j>|^2 $ ...
1
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1answer
27 views

Weak convergence in $BC(\mathbb R^+;X)$

I know that a sequence $u_n\in C([a,b];X)$ ($X$ is a Banach space) converges weakly to $u$ iff $\{u_n\}$ is bounded and $u_n(t)$ converges weakly to $u(t)$ for each $t\in [a,b]$. Does this hold if we ...
2
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1answer
43 views

Von Neumann algebra generated by a subalgebra

Let A be a C*-algebra of operators on a Hilbert space H. Show that if $A\subset K(H)$, then $\{A'\cap K(H)\}'\cap K(H) = A$ I do not have any idea about it. Please give me a hint. Thanks.
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1answer
31 views

A linear bijection to a Banach space must have bounded inverse

Suppose that $X$ and $Y$ are Banach spaces, and $D ⊂ X$ is a linear subspace, which may not be closed. Suppose that $T : D → Y$ has a closed graph (in $X\times Y$), and is $1-1$ and onto. If $D$ is ...
2
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1answer
42 views

How can I visualize the nuclear norm ball

I want to see what the unit nuclear norm ball looks like. So I think of matrices whose singular values add up to $1$. For simplicity, let's talk about symmetric, $2\times 2$ matrices (so that I can ...
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1answer
22 views

For a given Hilbert space find a tight frame with bound A

For a given Hilbert space and $A>0$ find a tight frame with bound A. I know that an ortho-basis is a tight frame with $A=1$. Can I extend this to any $A>0$ by just scaling the ortho-basis?
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2answers
33 views

Weak* boundedness and norm boundedness in the dual of a normed vector space

Let $X$ be a normed vector space over $\mathbb R$, not necessarily Banach. Let $X'$ denote the dual of $X$, that is, the set of all bounded, linear functionals on $X$: $$X'\equiv\{f:X\to\mathbb ...
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41 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
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64 views

Can you equip every vector space with a Hilbert space structure?

Suppose that we have a vector space $X$ over the field $\mathbb F \in \{ \mathbb R, \mathbb C \}$. Question: Does there exist a Hilbert space $\widehat X$ over $\mathbb F$ such that $\widehat X$, ...
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14 views

The Krein-Smulian Theorem and application

Let $H$ be a normed space and let $\tau$ be a locally convex topology on $H$ such that ball $H$ is $\tau$-compact. Show that there is a Banach space $Y$ such that $H$ is isometrically isomorphic to ...
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1answer
23 views

An Application: The Stone-Cech Compactification [duplicate]

If $X$ is completely regular, show that $X$ is open in $βΧ$ if and only if $X$ is locally compact.
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34 views

Theorem 3.6-4 in Erwin Kreyszig's Introductory Functional Analysis With Applications

Here's the statement of Theorem 3.6-4 in Erwin Kreyszig's Introductory Functional Analysis With Applications: Let $H$ be a Hilbert space. Then (a) If $H$ is separable, then every orthonormal set in ...
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1answer
23 views

Show that a nondegenerate *-Banach algebra is a C*-algebra

Takesaki in his operator theory says A C*-algebra $M$ of operators on Hilbert space $H$ means a nondegenerate ( $\text {cl} (MH) = H$) $*-$ subalgebra of $B(H)$ which is closed under the uniform ...
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1answer
38 views

$l^2(A)$ is unitarily isomorphic to $l^2(B)$ implies $card(A)=card(B)$

$l^2(A)$ is unitarily isomorphic to $l^2(B)$ implies $card(A)=card(B)$ How can we construct the map from A to B via the unitary map? Unitary map is a invertible map between Hilbert spaces that ...
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1answer
35 views

Mollifiers: Asymptotic Convergence vs. Mean Convergence

Problem Does asymptotic convergence imply mean convergence: ...
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1answer
69 views

what mean $\overline{\text{span}(e_1,…,e_n)}$?

In functional analysis, what mean $$\overline{\text{span}(e_1,...,e_n)} \ \ ?$$ is it the closure of $\text{span}(e_1,...,e_n)$ ? How does it work ? Even if it's that, I don't understand how ...
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38 views

Doubt Concerning the Definition of a Locally Convex Space Structure through Seminorms?

In the book Introduction to Functional Analysis written by A. E. Taylor there are the following theorems: Theorem 1. Suppose that $X$ is a linear space and that $\mathscr{U}$ is a nonempty family ...
3
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1answer
52 views

Sequence is not in any $\ell^p$ space

We know the sequence {$\frac{1}{ln(n)}$} such that $(n>=2)$ converges to $zero$ but is not in any $L_p$ space because of $$\sum_{n=2}^{\infty}\left|{\frac{1}{ln(n)}}\right|^p ={\infty}$$ for any ...
0
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1answer
25 views

$l^p$ is of first category in $l^q$ if $1 \leq p < q< \infty$

Let $1\leq p<q<+\infty$. Let $B_n=\{<x_k>\in l^q: \sum_k |x_k|^p\leq n\}$. Want to show: $B_n$ is closed and nowhere dense in $l^q$. Thus $l^p$ is of first category as a subset of $l^q$. ...
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66 views

Mollifiers: Derivative

This thread is meant as lemma for: Semigroups & Generators: Entire Elements: Construction Given a smooth mollifier: $\varphi\in\mathcal{L}(\mathbb{R}): \varphi'\in\mathcal{L}(\mathbb{R})$ Do ...