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

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5
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1answer
532 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 $U\subset\mathbb{R}^n$ ...
5
votes
2answers
283 views

Brownian Motion Covariance: max instead of min

It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
5
votes
1answer
168 views

Show $T$ is compact

$H$ and $K$ are Hilbert Spaces, $(u_n)$ and $(v_n)$ are sequences in $H$ and $K$ respectively. $\sum_{n=1}^{n=\infty} \|u_n\|\|v_n\| $ converges. $T\colon H\rightarrow K$ is defined by ...
5
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1answer
512 views

Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
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0answers
52 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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0answers
47 views

Two Hilbert spaces $V \subset H$, a basis for both spaces?

Let $V \subset H$ be a pair of Hilbert spaces (with different inner products). The embedding is continuous and dense, and both spaces are separable. Is it always the case that one can I find a ...
5
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1answer
164 views

Positive Operators: Definition?

Definitions Given an operator algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ with $1\in\mathcal{A}$ Consider selfadjoint operators $A=A^*\in\mathcal{A}$. Define positive elements by: ...
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0answers
172 views

Is there an orthonormal basis for $L_2[0,1]$ consisting of convex functions?

Is there an orthonormal basis $\{\phi_{\alpha}\}$ for the space $L_2[0,1]$ of square-integrable functions from $[0,1]$ to $\mathbb{R}$ such that every $\phi_{\alpha}$ is convex? Edit: A helpful ...
5
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1answer
215 views

Prove or disprove this argument

Let $L>0$ and let $\Omega$ be the set of all integrable functions from $[0,L]$ to $]0,+\infty[$. For all $\varphi, \psi \in \Omega$ define $\left \langle \varphi,\psi \right ...
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0answers
169 views

Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...
5
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1answer
204 views

the basis for the Sobolev space $H^1_0([0,1],\mathbb{R})$

According to the Sturm-Liouville theorem, for any continuous function $p\in\mathcal{C}^0([0,1],\mathbb{R})$, there is a Hilbert basis (normlised) $(\psi_n)_{n\geq1}$ of $L^2([0,1],\mathbb{R})$ such ...
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0answers
112 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
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votes
0answers
213 views

Hilbert spaces - equivalent norm

Let $H$ be a Hilbert space with a norm $\| \cdot \|_1$. Let $\| \cdot \|_2$ be another norm on $H$ which is equivalent with $\| \cdot \|_1$. It is easy to see that $(H, \| \cdot \|_2)$ is a Banach ...
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1answer
213 views

Conditions for the sequence being weakly convergent

Let $H=\ell_2$ be the Hilbert space of the square-summable sequences where $$ \langle x,y\rangle=\sum_{i=1}^{\infty}x_iy_i, \quad \|x\|=\sqrt{\langle x,x\rangle}. $$ Let $F: H\rightarrow H$ be an ...
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0answers
437 views

Sum of operator and adjoint is self-adjoint

In abstract Hodge theory there is the following lemma: Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
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6answers
656 views

Bounded operator that does not attain its norm

What is a bounded operator on a Hilbert space that does not attain its norm? An example in $L^2$ or $l^2$ would be preferred. All of the simple examples I have looked at (the identity operator, the ...
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3answers
1k views

Convergence in weak topology implies convergence in norm topology

In Hilbert space why does convergence in weak topology $x_n$ to $x$ imply that $x_n$ converges to $x$ in norm? Thank you very much for your answers. What if I put a condition on weak convergence ...
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2answers
417 views

Boundedness of operator on Hilbert space

I have the following question: let $\mathcal{H}$ be a Hilbert space and $\{\varphi_{i}\}_{i \in \mathbb{N}}$ be an orthonormal basis. Furthermore let $T: \mathcal{H} \rightarrow \mathcal{H}$ be an ...
4
votes
5answers
542 views

Why do bases of infinite dimensional spaces need to be orthonormal?

I asked this question following a discussion in my Mathematical Methods course and didn't get a satisfactory answer. If we have an infinite dimensional Hilbert space, why do we need an orthonormal ...
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votes
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
546 views

A counterexample to theorem about orthogonal projection

Can someone give me an example of noncomplete inner product space $H$, its closed linear subspace of $H_0$ and element $x\in H$ such that there is no orthogonal projection of $x$ on $H_0$. In other ...
4
votes
2answers
266 views

Is this operator bounded? Hilbert space projection

Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense in $H$. Let $b_n$ be an orthonormal basis for $H$ and an orthogonal basis for $V$. Define $$P_n:H \to ...
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2answers
96 views

A counterexample on the existence of some sequence in Hilbert space

I want to find a uniformly bounded sequence $\{x_n\}$ in $l^2(\mathbb{C})$ such that $x_n$ does not converge to zero in weak topology, i.e., $\exists ~y\in l^2(\mathbb{C}),$ such that $\langle y, ...
4
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2answers
154 views

Do all angles occur in Hilbert spaces?

Let $X$ be a Hilbert space with scalar product $(\cdot,\cdot)$. Then for two vectors $v,w$ of norm $1$, we can interpret $(v,w)$ as an angle, so that $(v,w)=\cos(\varphi)$ for a unique angle ...
4
votes
2answers
456 views

Paradox or Error: On the inclusion of dense subspaces into Hilbert spaces

the following observations are very simple, but I suppose they contain an error, which I haven't been able to find it so far. Maybe somebody can help how to fix it: Let $H$ be a Hilbert space, $U$ be ...
4
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2answers
177 views

One of these two operators is not invertible

I have a Hilbert space $H$ and a bounded self-adjoint operator $T$ on it with $||T||=1$. I've been trying to show that at least one of $I+T, I-T$ are not invertible, but I haven't been able to make ...
4
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1answer
213 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
4
votes
1answer
846 views

Hilbert-Schmidt Operator

We have just covered Hilbert-Schmidt operators in class (which I missed) and I am having a hard time getting my head around them. I know the definition: If $H$ is a Hilbert space and ...
4
votes
5answers
948 views

Subspaces of Hilbert Spaces of finite dimension

Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ ...
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4answers
2k views

Compact operator

If $H$ and $K$ are Hilbert spaces,show that if $T:H\longrightarrow K$ is a compact operator and $\{e_{n}\}$ is any orthonormal sequence in $H$ then $\|Te_{n}\|\to0$.Is the converse true? thanks.
4
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1answer
83 views

Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
4
votes
3answers
96 views

$A^2$ self-adjoint and Compact, prove $A$ has an eigenvalue

Suppose $H$ is a Hilbert space and $A \in L(H)$ is such that $A^2$ is compact and self-adjoint. Prove that $A$ has an eigenvalue. (Here $L(H)$ is the set of bounded linear operators on a Hilbert ...
4
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3answers
781 views

What is a Hilbert space?

I've just seen a question about Hilbert Subspaces. This made me wonder what a Hilbert space is. Can anyone explain in layman's terms?
4
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2answers
956 views

$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
4
votes
1answer
881 views

Formula for trace of compact operators on $L^2(\mathbb{R})$ given by integral kernels?

Given an appropriate function $K: \mathbb{R}^2 \to \mathbb{C}$, say continuous of compact support, we obtain a compact operator $T$ on the Hilbert space $L^2(\mathbb{R})$ by the formula $$ (T h)(t) = ...
4
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1answer
59 views

How to conclude $\Re $ is zero?

I'm in a Hilbert space $H$ and for $z,v, h \in H$ and $t \in \mathbb C$ I have $$ \|z\|^2 \leq \|h−(tv+y)\|^2 = \|z−tv\|^2 =\|z\|^2 −2\Re(t⟨v,z⟩)+|t|^2\|v\|^2$$ According to my notes it follows ...
4
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1answer
337 views

A linearly independent, countable dense subset of $l^2(\mathbb{N})$ [duplicate]

Possible Duplicate: Does there exist a linear independent and dense subset? I am looking for an example of a countable dense subset of the Hilbert space $l^2(\mathbb{N})$ consisting of ...
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4answers
1k views

Weak and pointwise convergence in a $L^2$ space

Let $I$ be a measured space (typically an interval of $\Bbb R$ with the Lebesgue measure), and let $(f_n)_n$ a sequence of function of $L^2(I)$. Assume that the sequence $(f_n)$ converge pointwise ...
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3answers
757 views

Separability of the space of bounded operators on a Hilbert space

Let $H$ be a (separable) infinite dimensional Hilbert space, and $B(H)$ the space of bounded operators on $H$. Is $B(H)$ separable in the operator norm topology? What about in the strong and weak ...
4
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1answer
174 views

Continuation of Linear Operator in Hilbert spaces

First of all, here is the assignment: Let $X$ be a Hilbert space over $\mathbb{C}$, $V \subseteq X$ be a closed subspace and $f \in L(V, \mathbb{C}) $ a linear continuous operator. Show that ...
4
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1answer
287 views

How to characterize self-adjoint operators in terms of orthogonal diagonalizability

Have a look at the following excerpt of Tosio Kato (taken from Zeidler Applied functional analysis vol. I): The fundamental quality required of operators representing physical quantities in ...
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1answer
79 views

Definitions of adjoints (functional analysis vs category thy)

If I have a linear operator $f$ on a Hilbert space, then I define the adjoint of $f$ to be $f^*$ where, $(fx,y)=(x,f^*y)$ for all $x,y$. I am confused because this definitions is very different to ...
4
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1answer
129 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
4
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1answer
275 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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1answer
93 views

Is this function positive?

I was wondering if: $$\int_0^1x(t)\int_0^tx(s)ds\ dt$$ is positive for a general $x\in L_2[0,1]$ . Can you help me with this?
4
votes
3answers
875 views

Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional

Suppose that we have a Hilbert Space $H$ and $M$ is a closed subspace of $H$. Let $T\colon H\rightarrow M$ be the orthogonal projection onto $M$. I have to show that $T$ is compact iff $M$ is finite ...
4
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1answer
867 views

Summation of inner products

I can't seem to find a way of asking a sub-question in relation to does linearity of inner product hold for infinite sum, which is in itself too generic a question for my purposes. Could someone ...
4
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1answer
74 views

Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta f(v) d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb ...
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3answers
81 views

Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
4
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1answer
143 views

Composition of two orthogonal projections

Let $V$ be a finite dimensional Euclidean space and let $W_1,W_2$ be two subspaces of $V$. Let $P_1,P_2$ denote the projections onto $W_1,W_2$ respectively. Is it true that the composition $P_1\circ ...