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

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29 views

Determining adjoint operator between spaces with different inner products

I'm given a space of $X \in R^{mxn}$ with inner product $\langle X_1, X_2 \rangle = tr(X_1^T X_2)$ and another space of random vectors $Y \in RV^m$ with inner product $\langle y_1,y_2 \rangle = ...
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0answers
45 views

Given two Hilbert Spaces, one is Isomorphic to a subspace of the other.

I suppose I'm doing something wrong as this solutions seems too simple. We know that each of $\mathcal{H}_1$, $\mathcal{H}_2$, has a maximal orthonormal basis. Suppose $\{u_\alpha\}_{\alpha \in ...
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0answers
14 views

Orthogonal complement of a single element of Hilbert space

Let $F = \{e\}$ be a set, where $\|e\| = 1$. My question is, what is $P_{F^\perp}h$ for all $h$ in a Hilbert space? Another question is, does $e$ all by itself, make up an ortonormal system?
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3answers
56 views

Having trouble showing a subspace of $\ell^2$ is closed.

Let $M =\{ (x_n)_{n \in \mathbb{N}} \in \ell^{2} \mid x_{2j} = 0, \text{ for all } j \in \mathbb{N}\}$. $M$ is a subspace of the Hilbert space $\ell^{2}$ and I'm supposed to show it's closed. ...
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93 views

Questions about Hilbert spaces, linear subspaces and orthonormal bases

I've been looking over some old assignments in my analysis course to get ready for my upcoming exam - I've just run into something that I have no idea how to solve, though, mainly because it looks ...
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17 views

The domain of a root of a self-adjoint operator associated with an interpolation space

We now that $V$, $H$ are separable Hilbert spaces such that $V$ is dense in $H$ and $V\hookrightarrow H$ continuous, by representation theorem exists $A: D(A)\subset V\rightarrow H$ self adjoint e ...
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2answers
47 views

Compatible Hilbert space subspaces - need help understanding a statement made in a book

A book I'm reading has the following in a section on lattices formed by subspaces of a Hilbert space : Two subspaces $M$ and $N$ are compatible if there exist three mutually disjoint subspaces ...
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5answers
468 views

What does it mean for two functions to be orthogonal? [duplicate]

When two finite dimensional vectors are orthogonal, i.e. perpendicular, their dot product is exactly zero, e.g. $$\mathbf{a}\cdot\mathbf{b}=a_1b_1+\cdots+a_nb_n=0.\tag{1}$$ When I studied functional ...
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1answer
61 views

Every complete orthonormal set in a Hilbert space $H$ is an orthonormal basis, if and only if $H$ is finite dimensional.

Show that any orthonormal set in a Hilbert space $H$ is linearly independent, and use this to show that $H$ is finite dimensional if and only if every complete orthonormal set is an orthonormal basis. ...
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1answer
48 views

Does a small perturbation of an orthonormal basis create strongly linearly independent vectors?

Let $e_1, e_2, \ldots$ be an orthonormal base in the separable Hilbert space $\mathcal{H}.$ Let $\psi_1^n, \psi_2^n, \ldots \in \mathcal{H}, n\in \mathbb{N}$ be vectors such that $\sup_{i} \| ...
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2answers
33 views

Every orthonormal set in a Hilbert space is contained in some complete orthonormal set.

Let $H$ be a Hilbert space. Show that every orthonormal set in $H$ is contained in some complete orthonormal set. I'm unable to start from any direction. Do I use the Gram-Schmidt orthogonalization ...
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0answers
16 views

Countable basis of differentiable function

If $$B = \left \{ \mathbf{v}_{k} \right \}_{k=0}^{\infty}$$ is a countable basis of a differentiable function $\mathbf{u}$ where $$ \mathbf{u} =\sum_{k=0}^{\infty}a_{i}\mathbf{v}_{k},$$ then show ...
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1answer
25 views

Trace Class: Decomposition

This is only Q&A. Preview Trace class operators decompose. So proofs reduce to Hilbert-Schmidt! Problem Given a Hilbert Space $\mathcal{H}$. For the trace class: ...
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1answer
35 views

Are Hilbert-Schmidt operators in non-separable Hilbert spaces compact?

The definition of Hilbert-Schmidt operator should still be valid even when the Hilbert space is not separable: If $e_i$ for $i\in I$ is an orthonormal basis for a Hilbert space, and ...
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1answer
26 views

Restrictions of function decomposition in $R^3$

I'm interested in the properties of countable basis functions that span functions living in $\Bbb R^3$. Can I represent a $L^2$ normalizable function that has a point divergence, (for example, ...
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1answer
42 views

Question about norm of linear operator's adjoint on a Hilbert space

Stein and Shakarchi state that for a bounded linear operator $P:H\to H$, $||P||=||P^*||$, where $||P||=\sup\{|(Pf,g)|:||f||\le 1,\ ||g||\le 1,\ f,g\in H\}$. Is it true that for any $f\in H$, ...
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2answers
34 views

Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff $(V,||\cdot||_2)$ is.

Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff $(V,||\cdot||_2)$ is. I really didn't get it. Of course both spaces are normed spaces but ...
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4answers
35 views

Is the zero vector always included in the orthogonal complement?

This might be a really silly question, but I'm currently studying for my upcoming analysis exam, and I can't quite find an answer to this. The reason I'm asking is actually, that I've been looking ...
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1answer
25 views

Is the closed span itself a hilbert space?

Let $(X_t)$ denote a process, where $X_t\in L^2(\Omega,F,P)$. Here, $L^2$ is a Hilbert space with inner product $\langle X,Y\rangle = E(XY)$. Maybe a stupid question but is the closed span $$ ...
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2answers
74 views

Is $B(H)$ sot separable

To prove that the unit ball of $B(H)$ is separable in strong operatior topology using the fact that $K(H)$ is separable and also is sot- dense in $B(H)$. I think we can conclude that $B(H)$ is also ...
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1answer
19 views

Norm of operator on $\ell_2$

I am having trouble computing the norm of the following operator: $$T:\ell_2 \to \ell_2$$ given by $$T(x_1, x_2, x_3, \dots) = \left(x_1, \frac{x_2}{2}, \frac{x_3}{3},\dots\right).$$
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1answer
40 views

Prove H (all m x n real matrices) is a Hilbert space

H consists of all m x n real matrices with addition and scalar multiplication defined as the usual corresponding operations with matrices, and with the inner product of two matrices A, B defined as ...
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1answer
58 views

Hilbert space has countable orthonormal basis iff it contains countable dense subset.

A Hilbert space $H$ has a countable orthonormal basis if and only if it contains a dense countable subset. I solved the "if" part. As for the "only if" part, when $\{e_n\}_{n\in \Bbb N}$ is a ...
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1answer
34 views

Conditional Expectation and Interpretation of Projection of $Y$ Onto $X$ vs. $Y$ onto $L^{2}(\sigma(X))$

Suppose $X,Y\in L^{2}(\Omega,\mathcal{F},\mathbb{P})$ are real-valued square-integrable random variables and that the joint density of $(X,Y)$ denoted by $f_{XY}(x,y)$ exists. Then as is well-known, ...
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1answer
25 views

Determining $x$ in a Hilbert space s.t $\langle x,e_n \rangle = 1/\sqrt{n}$, where $(e_n)_{n=1}^{\infty}$ is an orthonormal basis.

I have a quick question regarding Hilbert spaces: Given a Hilbert space, $H$, with an orthonormal basis, $(e_n)_{n=1}^{\infty}$, is it possible to determine $x \in H$ s.t $\langle ...
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1answer
33 views

Nested subspaces in a Hilbert space

Can one find an infinite decreasing chain $X_1\supset X_2\supset\dots$ of linear closed subspaces of $l_2$ such that $\bigcap X_i=\{0\}$, and for any $x\in l_2$ there exist $j$ such that $x\in ...
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1answer
18 views

Find the minimum coefficients in an inner product on L2(-1,1) using Legendre polynomials as orthonormal vectors.

Given that the orthonormal vectors in $L^{2}$(-1,1) obtained by applying the Gram-Schmidt process to 1,x,$x^{2}$ are scalar multiples of the first three Legendre polynomials: $P_{0}(x) = 1$, ...
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2answers
37 views

Is Eberlein-Smulian necessary to prove the fact that every bounded sequence of a Hilbert space has a weakly convergent subsequence?

The standard answer I have seen on stackexchange is something like this: Hilbert space is the dual of itself, so applying Banach-Alaoglu, we see that the bounded sequence is contained in a closed ball ...
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1answer
42 views

How could I find a non-closed linear subspace $X$ of $l^{2}$ , such that $ l^{2} \ne X + X^{\perp}?$

How could I find a non-closed linear subspace $X$ of $l^{2}$ , such that $l^{2} \ne X + X^{\perp} ?$
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1answer
26 views

Definition of a Hilbert basis

Given a Hilbert space $\cal H$, what criterion describes the property "$\cal B$ is a Hilbert basis for $\cal H$"? It would be even better if the definition can be stated in a way that mimics some ...
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1answer
66 views

How these norms are equivalent in an Hilbert space?

I have $H$ an Hilbert Space and $L:(H,\left\| \cdot \right\|_1) \rightarrow (H,\left\| \cdot \right\|_2)$ linear and bijective; here $\left<x,y\right>_2:=\left<Lx,Ly\right>_1$ and so ...
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0answers
20 views

Qubit in Hilbert space in its SU(2) representation.

A qubit is normally defined as $$|q\rangle = \binom{\alpha}{\beta}= \binom{\phi_0 e^{i\theta_0}}{\phi_1 e^{i\theta_1}}$$ in a two dimensional complex Hilbert Space where $\alpha$ and $\beta$ are ...
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1answer
54 views

Metric completion of polynomial function space

Take the vector space of all polynomial functions from $\Bbb R$ to $\Bbb R$ with an inner product $$\langle f,g \rangle = \int_{-\infty}^{\infty} f(x)g(x)\sigma(x)dx$$ where $\sigma$ is a positive, ...
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1answer
37 views

Find a function to minimize norm.

I have a problem I cannot find a solution to by myself. It goes like this: We have a Hilbert space spanned by the family of functions $\{\sin(x), \cos(x), \sin^2(x), \cos^2(x), \sin(2x)\}$. The scalar ...
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1answer
55 views

Trace Class: Counterexample

This is a real question! Given a Hilbert space $\mathcal{H}$. Denote trace class by:* $$\mathcal{B}_\textrm{Tr}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):\operatorname{Tr}|A|<\infty\}$$ Then ...
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1answer
32 views

Given a functional f, how do you find a v such that f(x) = (x,v)?

I have a functional $f: L^2[-1,1]\to\mathbb{C}$, how do I find a $v$ such that $f(x)=\langle x,v\rangle$ for all $x$? Since $L^2$ is a Hilbert space, I know the Riesz Representation theorem holds but ...
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62 views

What's the intuition behind the direct integral of a family of Hilbert spaces?

In order to understand better the mathematically rigorous version of Dirac's formalism in Quantum Mechanics I've been reading about direct integrals of Hilbert spaces, projector-valued measures and so ...
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1answer
70 views

Demonstration of Cauchy-Schwarz inequality using Minkovski inequality

I intend to demonstrate Cauchy-Schwarz inequality assuming that Minkovski inequality is true $(i)$. Instead of doing the usual proof I want to apply this second inequality somewhere in the resolution, ...
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0answers
27 views

For what operators $A$ on a Hilbert space is the identity operator in the closure of the similarity orbit of $A$?

For a bounded linear operator $A$ on a separable Hilbert space, the similarity orbit of $A$ is the set $S(A)=\{WAW^{-1}: W \text{ is invertible}\}$. I am wondering that if the identity operator $I$ is ...
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1answer
58 views

Example of complete orthonormal set in an inner product space whose span is not dense

Let $X$ an inner product space and $A$ be an orthonormal set and $\overline{Span(A)}$ = $X$ then $A$ is Complete. But the converse is not true until we consider $X$ as a Hilbert space. I am searching ...
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2answers
37 views

Does $A+A^\perp=\scr H$ in a Hilbert space imply $A$ is closed?

This is just the converse to the Hilbert projection theorem, which says that if $A$ is a closed subspace of a Hilbert space $\scr H$ then $A+A^\perp=\scr H$. If $A$ is a linear subspace of $\scr H$ ...
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1answer
31 views

Find the number of interior points of this subspace of $l^2$.

Consider the Hilbert Space $l^2$. Let $S=\{(x_1,x_2,\cdot\cdot\cdot)\in l^2:\sum\dfrac{x_n}{n}=0\}$. Then find the number of interior points of $S$. Let ...
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1answer
36 views

Dual Cone Containment

I have the following conjecture which, as of now, I can neither prove nor disprove: Let $H$ be a hilbert space, and let $C \subset H$ be a closed (convex) cone with the property that ...
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1answer
17 views

Obtain a formula for the projection $P_D$ and prove it using angle characterization

Consider the Hilbert product space $X\times X$. In this space, define the closed convex “diagonal” set by $$D := \{(x,x) | x \in X\}.$$ Obtain a formula for the projection $P_D$ and prove it. I am ...
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9 views

Cancellation Law in Adjoint Existence Proof

There's a proof for existence of the adjoint operator in our course notes which looks to me as if it violates the cancellation law dictating that if $AB = AC$ we can only be certain that $B = C$ if ...
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1answer
11 views

Given X and another Hilbert space Y , show that the product space X×Y is also a Hilbert space, with inner product ⟨(x, y), (u, v)⟩ = ⟨x, u⟩ + ⟨y, v⟩.

I am trying to prove the above question. I know that I need to some how show that it satisfies the following axioms (i) $\langle x,x\rangle \geq 0$ and $\langle x,x\rangle=0$ if and only if $x=0$, ...
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1answer
33 views

prove that $\ell_2$ Parallelepiped is compact (problem in notations for Nested sub-sequences!)

Let $A=\{(x_n)_{n\in\mathbb{N}}\;\big| \;|x_n|\leq\frac1n\quad ;n=1,2,\cdots\}$ be a subset of $\ell_2$, which is a Parallelepiped in $\ell_2$. I think I've stuck in a problem with notation ! ...
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1answer
20 views

Why is $\sqrt{T^*T}$ self-adjoint?

Let $T$ be a bounded linear operator over some Hilbert space $H$. Since $T^*T$ is a positive operator, it has a square root. Let $R=\sqrt{T^*T}$. Prove that $\forall u\in H, ||Ru||=||Tu||$. ...
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0answers
22 views

$A^{\perp\perp}=A$ for a closed subspace $A$ [duplicate]

This question is essentially the same as double Orthogonal complement is equal to topological closure, but I have fewer "known" results. (Specifically, the linked Q/A proves that ...
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0answers
47 views

To show Banach Operator Norm's equivalence

I am thinking the sentence which may be about the Banach algebra function: Let $A \in \mathcal{B}(H)$ where $H$ is Hilbert. \begin{equation} \| A \| = \sup_{ \| u \|, \| v \| \leq 1} | \langle Au, ...