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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66 views

Why Shape space is manifold?

In Shape analysis, often shape is considered as continuous parametrized closed curve and the shape space as Hilbert Riemannian manifold. Can any one help me to understand, why the shape space is ...
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3answers
37 views

ONB: Fourier Series

Given the Hilbert space $L^2([-\pi,\pi])$. Consider the orthonormal system: $$\mathcal{S}:=\{\frac{1}{\sqrt{2\pi}}e^{ikx}:k\in\mathbb{Z}\}$$ This is an ONB. How do I prove this? I guess, I could try ...
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17 views

Fuctions quasi-concave and quasi convex in Normed Spaces [closed]

Let $(V,+,R, \cdot)$ be a linear space, and let $D\subseteq V$ be a convex set. Let $f: D \to \mathbb{R}$ be a functional. We define $f$ to be quasi-convex if, for every $v,v'\in D$ and every ...
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1answer
13 views

Determining if a function is unitary.

Fix $\lambda\in \mathbb{R}$ with $\lambda \ne 0$ and define a mapping on $L^2(\mathbb{R})$ by: $(Uf)(t) = |\lambda|^{1/2}f(\lambda t)$. Show that U is unitary. Proof so far: To show U is unitary ...
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2answers
45 views

Spectrum bilateral shift

Let $U \in \mathbb{B}(\ell^2(\mathbb{Z}))$ be the bilateral shift. I want to shoow that $\sigma(U)=\mathbb{T}$. Using functional Calculus I have shown that $\sigma(U)\subseteq\mathbb{T}$. In order to ...
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3answers
38 views

Positive Linear Functional on $C[0,1]$

I have an exercise which seems to be missing some information. Or it could be that I really don't need that information at all. Please let me know what you think and give a solution if possible. Thank ...
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0answers
27 views

About measurability for operator-valued functions

Being $E_1$ and $E_2$ Banach spaces, and working in a finite measure space, I have the following two definitions of measurability for a function $f:\Omega\to\mathcal{L}(E_1,E_2)$: $\bullet$ I say a ...
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1answer
298 views

Prove $f(x)=1/x$ is not regulated.

Let $f:[0,1]→\mathbb{R}$ be given by $f(0)=0, f(x)=1/x$ for $x>0$. Prove that $f$ is not regulated. A function $f:[a,b]\to\Bbb R$ is a regulated function if given $\varepsilon>0$ there is a ...
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21 views

Calculus of variations with $y\ge 0$ constraint [closed]

In my problem of Calculus of variations $J(y)=\int_a^b L(x,y,y')dx=0$, I have : end point constraint; integral constraint; and $y\ge 0$. We are familiar with the first two constraints, but what ...
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34 views

Separable dual space implies existence of complete series

Let $\{x_n\}$ be a basis for a Banach space $X$ and let $\{f_n\}$ be the associated sequence of coefficient functionals. Prove or disprove: if $X^\ast$ is separable, then ${f_n}$ is complete ...
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2answers
40 views

Dirac delta of x

Just so there are no misunderstandings let me first ask whether it is true that: $$ \int_{-\infty}^{\infty}x\delta(x)\mathrm{d}x=0. $$ If that is not true, then I don't know anything about the Dirac ...
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34 views

component convergence implies convergence in $l_1$ [closed]

Show that if $x_n(j)\rightarrow x(j) $ for all $j $ $ \epsilon $ N with $x_n$ and $x$ belonging to $l_1$ then $x_n$ converges to $ x $ in $l_1$-norm ? (i.e. $\sum\limits_{j=1}^\infty |x_n (j) - ...
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0answers
26 views

Finding norm of functional coefficents in $C[a,b]$

Let $\{x_n\}$ denote Schauder’s basis for $C[a, b]$ and let $\{h_n\}$ be the associated sequence of coefficient functionals. Compute $||h_n||$. In young's book, a schauder basis for $C[a,b]$ ...
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1answer
24 views

Contraction mapping theorem with respect to the supremum norm.

Question: For continuous $f$ on the interval $[0,1]$, let $(Tf)(x) = x + \lambda \int_0^x (x-t)f(t)$. Find the range of values of $\lambda$ for which the transformation $T$ is a contraction with ...
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1answer
20 views

Norm of $A$ is zer0 when $\Bbb H$ is Complex Hiblert Space

If $\Bbb H$ is a $\Bbb C$-Hilbert space and $A\in \Bbb B(\Bbb H),$i.e. bounded linear operator on $\Bbb H$ such that $\langle Ah,h\rangle=0$ for all h in $\Bbb H$, then $A=0$ For the proof of this ...
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25 views

How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, ...
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1answer
47 views

Why $\mathbb R$ is not complete with the metric $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$?

Suppose $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$. Prove that $\mathbb R$ is not complete with this metric. This is exercise 12 from chapter 1 from Rudin's Functional Analysis. ...
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1answer
47 views
+50

Existence of a Frechet topology on the dual of a barreled space

I have a Hausdorff separated locally convex barreled space $(X,\tau)$ with topological dual $X^*$. My questions are: $Q_1$ Is there a topology $\tau^*$ on $X^*$ that is finer than the weak-star ...
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1answer
40 views

A question about orthogonal projection

Here is a quotation of a book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. (P245) Let $H$ be a separable Hilbert space and $\Omega\subset B(H)$ be a separable set and let ...
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31 views

Gradient of an Inner Product in a more general Vector Space

I was looking at the following question: Differentiating an Inner Product that was talking about the derivative of an inner product to be: $$ \frac{d}{dt} \langle f, g \rangle = \langle f(t), ...
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2answers
33 views

Harmonic functions and Holder's inequality

In the book Real and Complex Analysis by Rudin, it is given that by applying Holder's inequality to the $u(re^{i\theta})=\frac{1}{2\pi}\int_{-\pi}^\pi P_r(\theta-t)f(t)dt$ we get ...
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1answer
44 views

If a sequence converges weakly in a closed subspace $M$ of a Banach space, then the (strong) limit point is in $M$.

Let $M$ be a closed subspace of the Banach space $X$ and let $x_{n}\in M$ converge weakly to $x$. Show that $x \in M$. We use the following definition $x_n\rightharpoonup x$ in $X$, $x_n$ ...
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53 views

Complex Measures: Integrability

Approaches A complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to integrability as: $$f\in L(\mu)\iff f\in ...
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2answers
78 views

A question about a conditional expectation in C*-algebra

Let $\Gamma$ be a discrete group. Consider a conditional expectation $\Phi: B(l^{2}(\Gamma))\rightarrow l^{\infty}(\Gamma)$ defined by $$\Phi(T)=\sum_{g\in \Gamma}e_{g,g}Te_{g,g},$$ where $e_{g,g}$ is ...
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1answer
15 views

A question about close line segment in TVS.

Suppose $E$ a topological vector space,which need not be Hausdoff. $x,y\in E$ are different. How to prove the close line segment $\{\alpha x+(1-\alpha)y:\alpha\in[0,1]\}$ is closed. And should it be ...
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1answer
35 views

A question about linear functional on TVS

Let $E$ be topological vector space on field $\mathbb{R}$(or $\mathbb{C}$), which need not be Hausdoff. $f$ is a linear functional on $E$, and there are open set $U\subset E$ and $t\in \mathbb{R}$(or ...
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1answer
27 views

Radon-Nikodym: Integrability?

Let $\lambda:\Sigma\to\mathbb{R}_+$ and $\kappa:\Sigma\to\mathbb{R}_+$ be finite measures on $\Omega$. Then by Radon-Nikodym: $$\kappa(E)\leq L\cdot\lambda(E)\quad(\forall ...
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1answer
55 views

Does a compact operator always have a kernel?

I am sorry if this question is stupid..... I raise it when I read Lax's book Functional Analysis. We know that some integral operators are compact, for example an integral operator from $L^2[Y]$ to ...
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1answer
42 views

The proof of “Every quasidiagonal C*-algebra is stably finite”

Here is a quotation in a book "C*-algebras and finite-Dimensional Approximations" by Nate and Taka (P241). Recall that an isometry $s$ is called proper if $1-ss^{*}\neq0$ Definition 7.1.14 A ...
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0answers
26 views

p-direct sum and dual spaces

I have a problem with my assignment of Linear Analysis. It should be rather easy and straight-forward, but I have problems =(. Let E and F be normed spaces. For $p \in [1,\infty]$, define the ...
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1answer
21 views

How to get a smoothing operator from a rapid decreasing function?

From John Roe: Elliptic Operators, topology and asymptotic methods, page 82-83. Let $\mathcal{D}$ be a Dirac operator on the spin bundle $S$, then any section $s\in L^{2}(S)$ has a "Fourier ...
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15 views

Measurability of the composition of sections and probability measures

Let $X,Y,Z$ be Polish spaces endowed with their respective Borel sigma algebras, and $\Delta(Y),\Delta(Z)$ be sets of Borel probability measures on $Y$ and $Z$, endowed with the weak*-topologies (and ...
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1answer
27 views

Kolmogorov n-width of N+1 dimensional ball

For a normed linear space $\mathscr{X}$, let $\mathscr{A}\subset\mathscr{X}$ and $\mathscr{X}_N$ any $N$-dimensional subspace of $\mathscr{X}$. Define the $n$-width of $ \mathscr{A}$ in $\mathscr{X}$ ...
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27 views

Using function parameters as representation

I was wondering if there is some field of mathematics which analyzes situations where you use function partners as representations, e.g. for classification or regression. For example, let's say I ...
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14 views

Prove that every reproducing kernel is a positive matrix (and vice versa)

Let $\mathcal{H}$ be a functional hilbert space (defined over a set $S$) with a reproducing kernel K. Prove that: a) $K$ is a positive matrix means the queadtric form is positive, i.e ...
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0answers
24 views

Extension of measure beyond Jordan-measurable sets

I know that if a set $A$ is Jordan-measurable (according to the definition that can be found here in problem 8) with respect to measure $\mu$, then, for any measure $\tilde{\mu}$ that is an extension ...
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24 views

Extension of $\sigma$-additive measure beyond Lebesgue-measurable sets.

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа an unproven statement saying that the system of sets of $\sigma$-uniqueness for a $\sigma$-additive measure $m$ defined ...
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0answers
29 views

Parabolic holder norms

Let $Q=\Omega\times[0,T]$ be a cylinder with $\Omega$ bounded open set in $\mathbb{R}^N$. N.V.Krylov in "lectures on elliptic and parabolic equations in holder spaces" defines the parabolic holder ...
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0answers
10 views

A basic doubt on metrizable topological vector space [duplicate]

Suppose we have a topological vector space with a countable local base. Then it has a balanced $\{V_n\}$ such that $$V_{n+1} + V_{n+1} + V_{n+1} + V_{n+1} \subset V_n$$ Why ? I know that every ...
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0answers
28 views

Approximate unit of a separable C*-algebra

The following is a corollary of Takesaki's Operator Theory: My question: I do not know why the author says"there exists an n such that $||x(1-v_n)^\frac{1}{2}||<\epsilon$" . Please help me to ...
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2answers
89 views

Why are $L^p$ spaces for $p\not=1,2,\infty$ important?

$L^p$ spaces for arbitrary $1\le p\le\infty$ are a mainstay of basic functional analysis courses, but I've only seen them "in action" when $p$ is 1, 2, or $\infty$. Can anyone give an "elementary" ...
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1answer
25 views

Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence

Let $\{e_n\}$ be a complete orthonormal sequence in an Hilbert space $H$ and let $\{f_n\}$ be an arbitrary sequence of elements in $H$ s.t $$\sum_{n=1}^\infty\|f_n-e_n\|^2<1$$Show that ...
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4answers
75 views

What are non-obvious examples of measures obtained from linear functionals by the Riesz representation theorem?

In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued ...
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1answer
20 views

Proving for each seperatble hilbert space exist complete sequence

Let $H$ be a separable Hilbert Space. Prove that exists orthonormal complete sequence and give example for one non-orthonormal sequence. I thought taking orthonormal basis for $H$ denoted by ...
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0answers
23 views

Equality of extensions of Jordan measure

I find the following theorem in Kolmogorov-Fomin's Elements of the Theory of Functions and Functional Analysis (p. 280 of this Russian ed., p. 26 of 1963 Graylock English ed.): In order that two ...
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1answer
104 views

Is $\mathcal{B}(H)$ complemented in $\ell_\infty(I, H)$

Let $H$ be an infinite diensional Hilbert space. Consider unit ball of $H$ as index set, denote it by $I$, then we have an isometric embedding $$ ...
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1answer
42 views

Possible inconsistency in a subspace of $l^∞$

Is it possible to have a suspace of $l^∞$ in which every sequence has a finite number non-zero elements? if so, what would be the zero element of the space? This a problem of the book of Kreyszig ...
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1answer
31 views

An routine exercise about matrix norm

If $T_{n}\in M_{k(n)}(\mathbb{C})$ and $||T_{n}^{*}T_{n}-1_{k(n)}||\rightarrow0$, then $||T_{n}T_{n}^{*}-1_{k(n)}||\rightarrow 0$ too? (Here, $M_{k(n)}(\mathbb{C})$ denotes the $k(n) \times k(n)$ ...
3
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0answers
29 views

Norm of an element in a C*-algebra

The following is a part of a proof in Takasaki's Operator theory: Let $\epsilon>0$ and $A$ is a C*-algebra. For an $x\in A$, put $h=x^*x$ and $u_\epsilon=(h+\epsilon)^{-1}h$. We have then ...
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2answers
51 views

Sobolev space on union of two open sets

Let $\Omega_1,\Omega_2 \subset \mathbb{R}^n$ be open sets. Let $p \in [1,\infty]$. Let $u: \Omega_1 \cup \Omega_2 \to \mathbb{R}$ be a function such that $u|_{\Omega_1} \in W^{1,p}(\Omega_1)$ and ...