For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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1answer
210 views

Point spectrum in Hilbert spaces

Let $H$ be a Hilbert space and and $T\in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e the set of all eigenvalues of T) and let $E$ denote the spectral measure. I'm trying to ...
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1answer
122 views

proof for a basis in $L^2$

I know, correct me if I am wrong, that the functions $H_n(x)\exp(-x^2/2)$ form a complete basis in $L^2(\mathbb{R},dx)$, where $H_n(x)$ is the $n$th Hermite polynomial. This must be true also for ...
2
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0answers
53 views

Find a bounded function with a supporting point

Given, $g(Z)=\operatorname{tr}\phi(Z)$, where $\phi(Z)= Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right) Z$ where $Z$ is a real rectangular matrix with more rows than columns (tall and ...
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1answer
59 views

A basic result about operators on Hilbert space.

I am studying following result. Let $H$ and $K$ be Hilbert spaces and an operator $A \in B(H, K)$, which has closed range. The spaces $H$ and $K$ have the following orthogonal decompositions: $H = ...
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0answers
213 views

Orthogonal Projection on hilbert spaces

I found this exercise on a book, I guess it's not hard but don't know what to do. Let $H$ be a Hilbert space and let $P:H \rightarrow H$ be linear. If $P$ is a projection, i.e $P^2 =P$, and ...
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0answers
108 views

Completeness proof.

I'm getting stuck showing a space is a Hilbert space. For $\Omega$ an open, connected and bounded set in $\Bbb R^2$ with regular boundary $\partial \Omega$, let $V=\{v \in H^1(\Omega)\ ;\ ...
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1answer
452 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
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1answer
134 views

It is possible to generalize the “real” line to be able to embed $\omega_1$ or any uncountable ordinal into a finite segment of it?

This question is motivated from a previous question, but is in itself independent of it. So, I understand that it is not possible to embed $\omega_1$ or any uncountable ordinal into the real line, ...
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1answer
469 views

Dual Space as a Hilbert Space

I have this problem: Let $(X, \langle\cdot,\cdot\rangle)$ a Hilbert Space on $\mathbb{R}$ with Riez map $\mathcal{R}:X^{\prime}\rightarrow X$, define $[\cdot,\cdot]:X^{\prime}\times ...
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1answer
48 views

Question on notation: What does $0 \leq M \leq 1$ mean for a bounded operator $M$?

Let $\mathcal{H}$ be a Hilbert space and let $M\colon \mathcal{H} \rightarrow \mathcal{H}$ be bounded linear operator. I am working through a paper by Roger Godement from the 1950's. In one section ...
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4answers
2k views

Orthogonal complement of a Hilbert Space

I have this problem: Let $S$ be a subset of a Hilbert $H$ and let $M$ be the closed subspace generated by $S$. Show that $M^{\perp} = S^{\perp}$ $M = (S^{\perp})^{\perp}$ if $V$ is a subspace of ...
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2answers
705 views

Isomorphisms of inner-product spaces

I think I understand why all finite-dimensional vector spaces over a field $\mathbb{K}$ are isomorphic to $\mathbb{K}^n$. Any linear map $T: V \rightarrow W$ between finite-dimensional vector spaces ...
2
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1answer
197 views

Pde problem with Neumann BC's

Let $U \subset\mathbb{R}^n$ be a bounded open set with smooth boundary $\partial U$. Consider the Neumann boundary problem $$-\Delta u +u=f, \quad \left.\frac{\partial u}{\partial ...
2
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2answers
71 views

Inner product? Yes or no?

I define an "inner product" on $H_0^2(U)$ where $U \subset R^n$ is bounded open set: $$\langle u,v\rangle = \int_U \Delta u \Delta v dx.$$ I need this when trying to find a weak solution for my PDE ...
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0answers
142 views

Testing whether a finite measure is absolutely continuous with respect to Lebesgue measure using wavelets

I've been working through Fundamentals of Stochastic Filtering (Bain, Crisan) and am a little perplexed by the following (initially) seemingly straightforward exercise and its given solution. We are ...
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0answers
188 views

Inverse of Identity plus Volterra operator

consider the following operator or $L_2(0,1)$, $(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial. I am trying to construct the inverse of this ...
2
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1answer
635 views

PDE weak solution problem

My professor grades really strictly (details). I would be very happy if you could help me with this problem: Let $U \subset R^n$ be a bounded set. Consider $ \Delta^2 u = f$ on $U$ and ...
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1answer
45 views

Prove that $S$is a closed subspace of $H^2$ invariant under multiplication by $z$. Find the inner function $F$ such that $S=FH^2$

Let ${\alpha_n}$ be a sequence of points in the open unit disc such that $\sum(1-|\alpha_n|)<\infty$. Let $S$ be the set of all functions $f$ in $H^2$ spaces such that $f(\alpha_n)=f'(\alpha_n)=0$ ...
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0answers
92 views

A question on weakly convergence and norm convergence.

Let $2 \le p<\frac{2n}{n-2}$. Suppose that a sequence $\{u_k\}_k\subset H^1(\mathbb{R}^n)$ weakly converges to $u \in H^1(\mathbb{R}^n)$, and hence weakly converges to $u$ in $L^p(\mathbb{R}^n)$. ...
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398 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
6
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1answer
199 views

Measure on a separable Hilbert space

Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have $$ ...
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1answer
222 views

Is this sum of convex and concave functions a convex function?

Is this a convex function in $X$, where all the entries are real and $Y,\beta$ are constants where $X,Y$ are rectangular matrices and $\beta$ is a constant vector and $A,B$ are constant p.s.d ...
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1answer
118 views

Mutually orthogonal subspaces of $L^{2}(X,\mu)$

Let $(X,\mathcal{M},\mu)$ be a measure space. If $E\in\mathcal{M}$, we identify $L^{2}(E,\mu)$ with the subspace of $L^{2}(X,\mu)$ consisting of functions that vanish outside $E$. If $\{E_{n}\}$ is a ...
2
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1answer
100 views

Operator norm estimate

Let $H$ be a Hilbert space with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$. Furthermore, let $B\colon H\rightarrow C[a,b]$ be a bounded operator. According to the Riesz-Frechet theorem there is ...
3
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0answers
69 views

Estimate finite-rank operator

I have the the following problem. Let $H$ be a Hilbert space with orthonormal basis $(e_{j})_{j\in \mathbb{N}}$. Let $x\in [a,b]$, for all $h\in H$ $$ (Bh)(x) = \langle h,k_{x} \rangle,$$ with ...
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1answer
81 views

must a continuous function into hilbert space with all differences perpendicular be constant?

If $H$ is a Hilbert space and $f:[0,1]\rightarrow H$ is a continuous function such that $f(x)-f(y)\perp f(y)-f(z)$ whenever $x<y<z$, does $f$ have to be constant? By Pythagoras's theorem, the ...
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0answers
85 views

Confused about Bessel's inequality

I know that if $H$ is a Hilbert space and $(e_{j})_{j\in\mathbb{N}}$ is an orthonormal system in $H$ and $f\in H$. Then one has Bessel's inequality $$\sum_{j=1}^{\infty}|\langle f,e_{j}\rangle ...
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1answer
51 views

Particular series on Hilbert Space

Let $(H, \langle\cdot,\cdot\rangle)$ a Hilbert space and consider a sequence $\{x_n\}_{n\in\mathbb{N}}$ of $H$ such that: $$\langle x_n,x_m\rangle\ =\ \delta_{mn}\ =\ \left\{\begin{array}{ll}1, & ...
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2answers
188 views

Show that linear Operator on $\ell^2$ is unbounded

Currently, I am preparing for a next semester course and trying to figure out some basic concepts in functional analysis. Let $T:\mathcal{D}(T)\to \ell^2$ be defined by ...
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1answer
232 views

Is my statistician friend right/wrong on metric spaces and norms?

I was talking to a statistician friend of mine who said that instead of minimizing this function $\sum_{i,j}W_{ij}d_{ij}^2(X)$ over $X$ it would be better to solve an analogous minimization problem ...
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1answer
115 views

Weak to strong mapping

Let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be strongly monotone if there exists $\alpha>0$ such that $$ \langle F(u)-F(v), u-v\rangle\geq \alpha \|u-v\|^2, \quad ...
3
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1answer
379 views

From weak convergence to strong convergence

Let $H$ be a real Hilbert space and $F:H\rightarrow H$ be a mapping such that $$ (A)\qquad\qquad(u_n\rightharpoonup u_*, F(u_n)\rightarrow F(u_*))\; \Longrightarrow\;(u_n\rightarrow u_*) $$ We are ...
5
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1answer
341 views

Bounded operator and Compactness problem

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a ...
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3answers
66 views

Solving for positive semidefiniteness

Given a real matrix M, is there a matrix function f(M) such that $f(M)-M$ is guaranteed to be positive semidefinite, other than the idea of multiplying $M$ with its transpose and apart from the ...
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1answer
165 views

Norm of oblique projector and angle between subspaces

Take $V$ and $W$ closed subspaces of $H$ a Hilbert space with $V\oplus W=H$ (we'll assume this holds in the sequel, it may not be required everywhere but in the context of interest, it is always ...
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0answers
286 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
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2answers
119 views

Norm of a function, Smoothness Penalization

I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.
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1answer
124 views

Hilbert spaces and orthogonality sets

I need to prove if $X$ is a Hilbert space and $M$ and $N$ it's closed: $$ (M+N)^\perp=M^\perp\cap N^\perp $$ thanks
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1answer
45 views

Property of sequence of eigenvalues of an operator.

For a positive (self adjoint) operator $A$ with eigenvalues $\lambda_k$, is it possible to have the case when neither $\lambda_k\to \infty$ or $sup_k \lambda_k<\infty$ for example if a subsequence ...
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1answer
288 views

Proving that Uniform operator convergence implies strong operator convergence implies weak

Let $H$ be a Hilbert Space. Let $\{T_n\}$ be a sequence of bounded operators in $H$. I'm trying to prove that Uniform Operator Convergence implies Strong Operator Convergence implies Weak Operator ...
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1answer
170 views

Proving that $Range(T)$ and $Ker(T)$ are subspaces of a Hilbert Space, $H$

I will prove that for $T:H\rightarrow H$, A bounded linear operator on a Hilbert space H, $ker(T)$ and $range(T)$ are subspaces of $H$. Is this valid? I will appreciate any corrections that you all ...
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1answer
152 views

Prove or disprove that the given expression is “always” positive

I have previously asked a question and I tried to solve it by my own and it led to the question below: Prove or disprove that ...
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1answer
251 views

Norms involving positive operators

Let's say we have $A \leq B$. Is it then true that $||Ax|| \leq ||Bx||$ (where $x, A, B$ all belong to the same finite-dimensional Hilbert space $H$)?
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2answers
235 views

Help finding a norm and using the Riesz Representation Theorem.

Let $ {P_{2}}([0,1]) $ be the Hilbert space consisting of all polynomials of degree at most $ 2 $ (including the zero polynomial on $ [0,1] $) equipped with the inner product $ \displaystyle \langle ...
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1answer
66 views

Help with proving: If $X$ is a Hilbert $A$-$B$-module, then $ \| _A \langle x,x \rangle \| = \| \langle x,x \rangle _B \| $ for all $x\in X $.

Sorry, I posted a related question last week on here, but I'm still having trouble and this is a little different, I hope it's OK. Thank you! ( proof that this is an isometric map (on a $C^*$-module) ...
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1answer
56 views

Hilbert, Banach and isomorphism

I want to show that if linear mapping $L:B_1\rightarrow B_2$ is isomorphism of Banach space and $\|L(x)\|_{B_1} =\|x\|_{B_2} $ (surjective and isometry) so it consist that $L$ is isomorphism of ...
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1answer
25 views

Confusion related to reproducing kernels

I was reading this paper and I came across Reproducing Kernel Hilbert Space. I tried to read some references related to it. However, I couldn't understand much. I didn't get why they are called ...
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0answers
151 views

When are two operators simultaneously diagonalizable?

I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...
2
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1answer
204 views

Invertible operator norm bound

Let $H$ be a Hilbert space and that $X$ are bounded. Suppose $X$ is self-adjoint. Show that $Y=X+iI$ is invertible and the inverse $Y^{-1}$ has the norm $\lVert Y^{-1} \rVert \le 1$. I can prove $Y$ ...
2
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1answer
46 views

On the existence of a bounded linear functional

Let $\mathcal{H}$ be a Hilbert space. By the Riesz Representation Theorem, we have that any bounded linear $\psi \in \mathcal{H}^{*}$ is of the form $\psi(h) = \langle h, g \rangle$ for some $g \in ...