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

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118 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
197 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
151 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
131 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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98 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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1answer
180 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
286 views

Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
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292 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
456 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 ...
4
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2answers
277 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 ...
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2answers
388 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 ...
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3answers
371 views

Soft Question Hilbert Space Geometry

Just a quick question about the geometry of Hilbert spaces from an intuitive standpoint. Maybe just assuming we're working with $L^2$ would simplify the situation. Basically, in something like ...
4
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2answers
150 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
91 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
136 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 ...
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3answers
405 views

$f$ an isometry from a hilbert space $H$ to itself such that $f(0)=0$ then $f$ linear.

This question was on an exam and I am not sure how to answer it. I mostly tried writing zero in different ways and tried lots of algebra to get something out. I also tried to use the fact that $H$ is ...
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2answers
378 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 ...
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1answer
519 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
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3answers
75 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 ...
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1answer
155 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 $\dim(H)>1$?
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5answers
707 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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1answer
57 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 ...
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1answer
271 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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1answer
246 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
271 views

Uniform mean ergodic theorem

I'm working in Einsiedler and Ward's book on Ergodic Theory and in Exercise 2.5.4 they want to prove the following $$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$ ...
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1answer
67 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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2answers
94 views

Norm of a $2\times 2$ matrix as a Hilbert space operator

Work in the Hilbert space $\mathbb C^2$. Let $$A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ be a matrix with entries in $\mathbb C$, and let $A$ ...
4
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1answer
228 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
88 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?
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4answers
643 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 ...
4
votes
1answer
152 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
votes
1answer
381 views

Reproducing kernel Hilbert spaces and the isomorphism theorem

A reproducing kernel Hilbert space is a Hilbert space in which the evaluation functional $L_x : f \rightarrow f(x)$ is continuous. By continuity, the Riesz representation theorem says that this ...
4
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1answer
508 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 ...
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2answers
597 views

Example that in a normed space, weak convergence does not implies strong convergence.

The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following definitions. A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is ...
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1answer
121 views

Dual and adjoint operator

Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual ...
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2answers
136 views

Hermitian matrices and great circles

I am considering parameterised curves in an $n$-dimensional complex vector space, given by the solution to the discrete Schrödinger equation: $$ |\psi\rangle(t) = e^{-iHt}|\psi_0\rangle, $$ Where $H$ ...
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2answers
112 views

Showing $\mathcal{H}$ is a hilbert space.

So this is an early exercise in Conway's A Course In Functional Analysis. I'm trying to get to grips with this upto open mapping and closed graph to see if I want to do any more functional analysis. ...
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2answers
163 views

a trace class operator problem

Could someone help me with this Prove that If $A$ and $B$ are positive trace class operators on a Hilbert space, then so is $A^zB^{(1-z)}$ for a complex number $z$ such that $0 <Re(z)< 1$. An ...
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2answers
398 views

Norm inequality for sum and difference of positive-definite matrices

If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear ...
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1answer
277 views

Bounded operator from a Hilbert space to $\ell^1$ is compact

Let $H$ be any Hilbert space. How can we prove that any bounded linear operator $T\colon H \to \ell^1$ is compact? If we use the fact that the space $\ell^1$ has Schur property (norm and weak ...
4
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1answer
94 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
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2answers
85 views

(From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$

Lang $SL_2$ p. 13 :Let $\{\phi_i\}$, $\{\psi_i\}$ be orthonormal bases for $L^2(X)$ and $L^2(Y)$ respectively. Let $$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$ Then $\{\theta_{ij}\}$ is an ...
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1answer
64 views

How can we pick $f \in C(0,T;H)$ with $f(T) =0$ and $f(0) = h$, where $h$ is arbitrary?

Let $C(0,T;H)$ be the space of continuous functions $f:[0,T]\to H$ where $H$ is Hilbert. For every $h \in H$, why is it possible to pick a function $f \in C(0,T;H)$ such that $f(0) = h$ and $f(T) = ...
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votes
2answers
438 views

Tensor product of operators

We know that if $T_1$ is a linear bounded operator on a Hilbert space $H_1$ and $T_2$ is a linear bounded operator on a Hilbert space $H_2$ there exists a unique linear bounded operator $T$ on $H_1 ...
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2answers
129 views

Is duality an exact functor on Banach spaces or Hilbert spaces?

Let $V,V',V''$ and $W$ be vector spaces over $k$. Then, it is known that $\operatorname{Hom}(\cdot,V)$ is a contravariant exact functor, i.e. for each exact sequence $0\to V'\to V\to V'' \to 0$, and ...
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2answers
173 views

Trying to understand Hilbert Spaces…

I am trying to get a hold on Hilbert Spaces, but I am having difficulties combinging various definitions. I have looked it up on wikipedia and wolfram, there it states something like "A Hilbert ...
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1answer
133 views

Convex subset of Hilbert space as intersection of closed balls

How does one prove that any closed, convex, and bounded subset of a Hilbert space is the intersection of the closed balls that contain it?
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2answers
742 views

What is the difference between a complete orthonormal set and an orthonormal basis in a Hilbert space

I don't know if it's true but I think in a finite dimensional veccter space, an orthonormal set which is complete becomes a basis. But in a Hilbert space, the books say that the set of finite linear ...
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1answer
186 views

Show that if the Riesz map is surjective on $H$, then $H$ is a Hilbert space

Let $H$ be a vector space equipped with an inner product $(\cdot, \cdot)$ and $f:H\to H',\ f(x)=(\cdot,x)$ surjective. Now, why $H$ is a Hilbert space? The other direction is clear by Riesz' ...
4
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1answer
103 views

Is it possible to 'approximate' compact, convex sets in $\ell^2$ by the Hilbert cube

Define $H=\{(x_n)_n\in\ell^2:|x_n|\le \frac1n, n\in\mathbf N\}\subset\ell^2$. This set is known as the Hilbert cube and it is well-known that $H$ is compact, convex and non-empty. Let ...