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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Constructing an L2 function from an entire function bounded on R

I have an entire function $f(z)$ of exponential type $\tau\geq0$ that is bounded on $\mathbb{R}$ and zero at every member of the complex sequence $\{\lambda_n\}$. What I want is an entire function of ...
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44 views

average of a bounded convex set

Suppose $X$ is a bounded convex set. We know that the average of any $n$ points of $X$, belongs to it, i.e. if $x_1, x_2, . . . , x_n \in X$ then $\frac{x_1+x_2+\cdots +x_n}{n}\in X$. How can we ...
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59 views

Identity plus finite rank has index $0$

I'm supposed to prove the strong Fredholm alternative in the form $$\text{Ind}(1-K)=0$$ for any compact operator $K:H\to H$ where $H$ is a Hilbert space and $$\text{Ind}(T):=\text{dim Ker }T+\text{dim ...
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50 views

unconditional basis inequality

I'm reading a paper and having some trouble with a certain inequality. Let $W$, $X$, and $Y$ be Banach spaces, with $(x_n)$ a normalized basic sequence in $X$ and $(w_n)$ a normalized unconditional ...
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33 views

Find a example in $L^2$, nonempty closed but contains no element of smallest norm. [duplicate]

As we know in Hilbert Space, a nonempty closed convex set always contains a element of smallest norm. I wanna find a example to show the convexity in some way is necessary. No loss of generosity, for ...
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34 views

Find the completeness radius of the prime numbers

As the title says, I'm trying to find the completeness radius of $\{2,3,5,7,11,\ldots\}$. The completeness radius of a sequence $\Lambda=\{\lambda_n\}$ is $R(\Lambda)=\sup\{A~|~\{e^{i\lambda_n ...
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58 views

Using Weyl sequences to prove relation between quadratic form and spectral radius

I know that the formula $$\lVert A\lVert=\sup_{\lVert x\lVert=1} \langle x,Ax\rangle$$ holds true for self adjoint operators. While reading Teschl's book I saw a comment that on can prove this formula ...
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25 views

the set of points that are annhilated by a subset $N\subset X'$

Let $X$ be a normed vector space over $\mathbb k$ ($\mathbb k=\mathbb R$ or $\mathbb C$). Let's consider $X'$ the set of continuous linear functionals $f:X\to \mathbb k$ called the dual of $X$. We ...
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54 views

Uniqueness of a weak derivative

Let $f\in \mathrm{L}^2[a,b]$. As usually $f'$ is the so called weak derivative of $f$ if $\forall \phi \in C_c^{\infty}(a,b)$ $\int_a^b f'\phi dt=-\int_a^bf \phi' dt$. Is it reasonably to think that ...
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29 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
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69 views

Topology of Convergence

I am having some difficulties in understanding the concept of topology induced by convergence? especially how the weak convergence induces weak topology? Does anybody know a good reference which ...
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46 views

An equicontinous headache (problem)

There's this family of functions: $$\mathcal H=\{ f_k\in\mathcal C^0([-1,1],\Bbb R) : f_k (x)=\begin{cases} -1 \;\text{if $x\in[-1,-\frac1k]$} \\ kx \;\text{if $x\in[-\frac1k,\frac1k]$} \\ 1 ...
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100 views

Borel Measure is Outer Regular iff it is inner regular?

I have a doubt on whether this theorem is true: Let $(X,\Gamma)$ be a compact space. Then a Borel measure $\mu$ on $\mathbb{B}(X)$ is outer regular iff it is inner regular. Can anyone shed some light ...
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43 views

Bochner integration and the associated notion of measurability

In http://en.wikipedia.org/wiki/Bochner_integral a notion of measurability is discussed that depends on the measure $\mu$. Usually measurability does not depend on having a measure anyway. Is this ...
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33 views

Composition of analytic functions is analytic in a general setting, and are they continuous?

Regarding the notion of analyticity discussed in this setting: A possible equivalence for holomorphicity I wonder if this is truly the correct definition (even though it is from Dunford-Schwarz) An ...
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43 views

Is this operator closed?

Consider the linear operator $H$ with domain $D(H) = S(\mathbb R)\subset L^2(\mathbb R)$, where $S(\mathbb R)$ is Schwartz space, defined by \begin{align} H\psi(x) = -ix^3\frac{d\psi}{dx}(x) -i ...
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2answers
62 views

Orthogonal completion in nonhilbert spaces [duplicate]

Let $X$ be some Hilbert space. There is the widely known theorem in functional analysis which states that for each closed subspace $H\subset X$ we have $H\bigoplus H^{\perp}=X$. Now we do not suppose ...
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61 views

Polish space: C(X,Y) when?

For which X,Y is the space of continuous functions $C(X,Y)$ from $X$ to $Y$ with the supremum norm a polish space and why? Is it enough if $X$ is compact and $Y$ is Polish? why?
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45 views

A question about bounded operators on banach space [duplicate]

Let $L(X)$ denotes the Banach algebra of all bounded linear operators acting on a Banach space $X$. And $T$ is not invertible. Can we find a invertilbe bounded operator series $\{T_{n}\}$ such that ...
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70 views

Sequences of 'Rayleigh-like quotients' and their minima for a symmetric positive semi-definite matrix

Let $A$ be an $N\times N$ symmetric positive semi-definite matrix with eigenvalues $0 \leq \lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_N$ and corresponding eigenvectors $u_1, u_2, \dots, u_N$. ...
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233 views

An Introduction to Functional Analysis: Prerequisites

There is a Coursera course starting this January called "An Introduction to Functional Analysis". I am interested in taking this course, but I'm not sure whether or not the material will be beyond me ...
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53 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
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67 views

Bounded linear mappings in Hilbert space preserve orthogonality?

My question is the title of this thread! Assume we have a bounded, linear mapping $A:H\to H$ where $(H,\langle\cdot,\cdot\rangle)$ is a Hilbert space, and two non-zero elements that are orthogonal, ...
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32 views

Open maping theorem. Completeness assumption are important [duplicate]

The open maping theorem between banach spaces says. Let $T:X\to Y$ be a linear,continuous and surjective map between the banach spaces $X,Y$ then $T$ is an open map. I need examples to show that the ...
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64 views

Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is \begin{equation} \omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})} \end{equation} where ...
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66 views

Fourier transform of displaced airy function

I need to find the fourier transform of displaced airy function.The function is $ψ_n(ξ) = N_n \text{Ai}(ξ − ξ_n)$, where $ξ=x/x_0$, $x_0=(1/2)^{1/3}$, $ξ_n = (3\pi/2)(n − 1/4)^{2/3}$ and $N_n$ is ...
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174 views

Minimal distance between quadratic function and point

I have a function or line $R\rightarrow R^n$ $$ y_i = f(x) = {-b_i \pm \sqrt{-4 \cdot a_i\cdot c_i + b_i^2+4\cdot c_i \cdot x}\over 2 \cdot c_i}$$ where the parameter vectors $\mathbf{a}, ...
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57 views

Codimension in X, given the codimension in a subspace of X

Let $X$ be a normed space with infinite dimension, let $M$ be a dense subspace of $X$ with codimension $k$ and let $N$ be a dense subspace of $M$ with codimension $l$ in $M$. The codimension of $N$ ...
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95 views

Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
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39 views

Existence results for this ODE? (periodic)

Are there any existence/uniqueness results for solutions to the ODE $$y'(t) = f(y(t),t)$$ $$y(0) = y(T)$$ on the time interval $[0,T]$ where $f$ is Caretheodory and $T$-periodic in $t$. I am looking ...
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Excercise 1.13 in Brezis's Functional Analysis

This is the Excercise 1.13 in Brezis's Functional Analysis Let $E=\mathbb{R}^n$ and let $$P=\{x\in\mathbb{R}^n;x_i\geq 0\ \forall i=1,2,...,n\}$$ Let $M$ be a linear subspace of $E$ such that ...
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82 views

Sturm-Liouville Eigenvalues

Consider Sturm-Liouville endpoint problems of the form $y''+\lambda y=0$ with the usual endpoint conditions. $c_1y(a)+c_2y'(a)=0$, $d_1y(b)+d_2y'(b)=0$. Here $(c_1,c_2) \neq \vec{0}$ and $(d_1,d_2) ...
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59 views

Sufficient and necessary condition for compact ellipsoids in $l_2$

Another fun problem from functional analysis that I am having issues with. I have fought long enough and would like to offer this to the community. For a sequence $\mathbb{R}\ni a_i>0$ consider a ...
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123 views

Eigenvalues of differential operator

If $L : C^2[a,b] \rightarrow C^0[a,b] $ is s.t. $L y(t) = \ddot y(t) +p \dot y + q y(t) $ and $L$ is invertible then $L^{-1}$ has at most countable eigenvalues and they accumulate in $0$. Why ...
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Show that $(D_t t)$ is an isomorphism.

Let $B$ be a Banach space, $\epsilon > 0$, and $$C_1^0 ([-\epsilon,\epsilon],B) = \{u \in C^0([-\epsilon,\epsilon],B) : tu \in C^1([-\epsilon,\epsilon],B)\}.$$ Denote $\partial/\partial t$ by ...
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76 views

Does my fixed point proof hold?

I have am looking for existence of a fixed point for an operator that I have. I already looked at some related fixed point theorems such as Schrauder's and Rothe's. But most of them seem to require ...
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145 views

Green' s function for harmonic oscillator

Does someone know how to get a solution of differential equation for Green's function $(-d^2/dt^2 + \omega^2) G(t, s) = \delta(t-s) $? There is a periodicity of G, actually $\Delta (t-s) = G(t,s)$ ...
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93 views

Equivalent statements about linear functionals in connection with the topology which we can obtain from seminorms

I have got a question about the topology we can get from out a separeting family of seminorms (to make a topological vector space from out an arbitrary vector space). There exists the following ...
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75 views

Representation of subspaces as complemented subspaces

Let $X$ be any separable Banach space. The Banach-Mazur theorem states (astonishingly) that $X$ is isometric to a closed subspace of $C(\Delta)$, the space of continuous functions on the cantor set ...
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Help with limit $\displaystyle\lim_{t\to 0^{+ }}\sup_{x\in[0, \infty)} |e^{-t^2-2tx}f(x+t)-f(x)|=0$..

can anyone help me showing $$\displaystyle\lim_{t\to 0^{+ }}\sup_{x\in[0, \infty)} |e^{-t^2-2tx}f(x+t)-f(x)|=0,$$ where $\displaystyle f\in C_0([0, \infty))=\{f\in C([0, \infty)): \lim_{x\to \infty} ...
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When does Gelfand Naimark Theorem Hold?

I was going through the proof of the Gelfand Naimark Theorem for the Unital Commutative Banach Algebras. In proving that each character has norm equal to $1$, we used the fact that $||e|| = 1$ where ...
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52 views

Intuition concerning Schwartz kernels of Operators

Consider a (for example differential) operator $A$ acting on an appropriate function space over a smooth compact manifold without boundary. Using the Schwartz kernel $K(x,y)dy$ of the operator, its ...
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63 views

Transpose of the Hilbert-Schmidt operator

Let $X = L^2(\Omega)$, $\Omega \subset \mathbb{R}^N$ be an open set (or a $\sigma$-finite measure space), $B \in L^2( \Omega \times \Omega)$. Then the Hilbert-Schmidt operator $T \in \mathcal L(X)$ ...
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119 views

an application of Hahn-Banach theorem

Let $M$ be a subspace of $L^1(\mu)$.Construct a bounded linear functional on $M$ such that there are two (hence infinite) different linear extensions preserving norm to $L^1(\mu)$. Someone outlined me ...
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51 views

positive definite operator and unique solution

In the page 32/39 of this paper http://archive.numdam.org/article/M2AN_1981__15_1_41_0.pdf they have the following equation: $T_h u_{h,t}+u_h=T_h f$ and say: "Since $T_h$ is definite positive (...) ...
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54 views

Does existence of weak spatial derivative imply existence of classical time derivative in this situation?

Let $f(x_1,...,x_n,t)$ be a function, where $(x_1,...,x_n) \in \mathbb{R}^n$ and $t \in [0,T].$ Denote by $f_{x_i}$ the weak (partial) derivative of $f$ wrt. $x_i.$ Is it possible for ...
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142 views

Are there differences between weak convergence and convergence in the weak topology?

In some book I found incongruences in notation that are misleading, so the following question arises: are weak convergence and convergence in the weak topology always the same thing? (the same for ...
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113 views

closed subspaces of locally convex inductive limits

Let $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ be an inductive sequence of LCTV spaces. A locally convex inductive limit of $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ is it's ...
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1answer
84 views

Riemann Stieltjes integral on $[0,1]$

I am looking for a hint or feedback on what I've already done, not a full solution So say we have the function defined on the unit interval by: $$ \alpha\left(\frac{1}{2}\right) =1, \alpha(t)=0, ...
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44 views

How to show that entries of this matrix are in $L^\infty(0,T)$?

I have a problem. Let $A(t)$ be a $n \times n$ matrix for each $t \in [0,b]$ with the property for all vectors $x$ that $$x^TA(t)x \geq C|x|^2$$ where $C$ doesn't depend on $t$. Can I use this fact ...