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

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2
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1answer
50 views

How to prove, that the ordering on positive bounded operators agrees with ordering of their ranges?

Hypothesis: Assume, that $A$ and $B$ are positive bounded operators (on some Hilbert space $H$) and $A\geq B \geq 0$. Then ${\rm range}(A) \supset {\rm range}(B)$. The textbook "$C^*$-algebras by ...
0
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1answer
59 views

Find minimal $\alpha_3$ such that $u\in H^3(\Omega)$ and $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$

My instructor presented me the quiz below but forgot to define key terms such as minimality and $H^3$. Quiz Let $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$. Find the minimal $\alpha_3$ such that $u\in ...
3
votes
1answer
96 views

Is $B - B'$ self-adjoint provided $B,B'$ are positive operators?

If I have two positive operators $B,B'$ on an arbitrary Hilbert space $\mathcal{H}$ not necessarily over $\mathbb{C}$, how do I know that $B - B'$ is self adjoint? EDIT: Reed and Simon define ...
3
votes
1answer
53 views

When does an operator commute with another operator given by a series?

Suppose $B$ is a bounded operator on some Hilbert space $\mathcal{H}$, given by a series of the form $$ B = I + \sum^\infty_{k = 1} c_k(I - A)^k $$ where $A$ is a given bounded operator on ...
1
vote
1answer
176 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 ...
3
votes
1answer
55 views

Proving $||A||=||A^{*}||=||AA^{*}||^{1/2}$

I am studying functional analysis, In the lecture notes I saw the claim: Let $A\in L(\mathcal{H})$ where $\mathcal{H}$is a Hilbert space. then $$ ||A||=||A^{*}||=||AA^{*}||^{1/2} $$ There is ...
0
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2answers
108 views

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$ I need to know whether it is self adjoint and unitary operator given that $x_i\in\mathbb C$ I am not able to do it please tell me how ...
3
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2answers
255 views

Is it a unitary, self adjoint and normal operator?

Let $A\colon H\to H$ be a bounded linear operator on a complex Hilbert space such that $\|Ax\|=\|A^*x\|\forall x$, given that there is a nonzero $x$ for which $A^*x=(2+3i)x$. Then I need to know ...
2
votes
1answer
48 views

Show that $v \in H^1(\Omega)$ if $v\in C^0(\Omega)$ and $v|_{\Omega_j} \in H^1(\Omega_j)$

Let $\Omega$ be an open set in $\Bbb R^d$. Let $\{\Omega_j\}_{j=1}^{N}$ be a fi nite collection of open disjoint subsets of $\Omega$ such that $\overline\Omega=\cup_{j=1}^{N}\overline\Omega_j$. ...
5
votes
0answers
221 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 ...
1
vote
0answers
68 views

How many projectors do two commuting self-adjoints have in their common spectral decomposition?

If $A$ and $B$ are two commuting observables on a Hilbert space of dimension $n$ say. So, $$A = \sum_{j \leq a} \lambda_j P_j $$ $$B = \sum_{i \leq b} \mu_i Q_i $$ $$I_n = \sum_{i \leq b} Q_j = ...
1
vote
1answer
189 views

l2 norms, rapidly decreasing functions and fourier transforms

Let $f\colon \mathbb{R} \to \mathbb{C}$ be a rapidly decreasing (rd) function. Let $\mathcal{F}(f)$ be the Fourier transform of $f$. It is known that 1) $\| \mathcal{F}(f) \|_2 = \| f \|_2 $ ...
1
vote
1answer
87 views

closedness a subset of a Hilbert space

Let $H$ be a Hilbert space that admits a countable orthonormal basis $\{e_i\}$. I know this means that $H$ is separable and so is $S$ (as a subset of it, defined below). Show that $S$ is a closed ...
0
votes
1answer
66 views

Convergence of an operator in norm

Let $H$ be a Hilbert space and assume we have three converging sequences: $u_n\rightarrow u$ in $H$, $v_n\rightarrow v$ in $H$ and $\lambda_n\rightarrow \lambda$ in $\mathbb{C}$. I would like to prove ...
0
votes
3answers
391 views

Showing that inner product of two vectors is the limit of the inner products

How can you show that $$(a,b) = (\sum_{i=1}^\infty a_i \phi_i,\sum_{i=1}^\infty b_i \phi_i) = \lim_{N\to \infty}(\sum_{i=1}^N a_i \phi_i,\sum_{i=1}^N b_i \phi_i)$$ where $\phi_i$ is an orthonormal ...
3
votes
1answer
320 views

Equivalent norm in sobolev space H^2

I consider space $H^{2}(0,a)=\{ f\in L^{2}(0,a): f',f''\in L^{2}(0,a) \}$ I define norm $\Vert w \Vert_{H^{2}}:=b\Vert w''\Vert_{L^{2}}$, where b is positive constant. I couldn't proof that it is ...
4
votes
1answer
65 views

Norms arising from all representations of *-algebras

It is common that in order to obtain a $C^*$-algebra from a $^*$-algebra $A$ one defines a norm on $A$ by $$\|x\|=\sup\{\|\pi(x)\|\,|\,\pi\ \text{is a }^*\text{-representation of }A\}.$$ However, I ...
4
votes
1answer
280 views

How to determine a operator norm?

How to solve following: In Hilbert space $W_2^1=\{f:[0,1]\rightarrow \mathbb{C}|f\in AC[0,1], f'\in L^2[0,1]\}$ with scalar product $(f,g)=\int_0^1 f\overline{g}dx+\int_0^1 f'\overline{g'}dx$ is ...
1
vote
1answer
82 views

Problems with understanding the proof for existence of projections to a close convex set on a Hilbert space

In the setting of an introduction to functional analysis course, I have read the following statement: Let $H$ be a Hilbert space and let $A\subseteq H$ be a closed convex set. Then there exist a ...
0
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1answer
309 views

A Riesz representation theorem without coercivity

Let $b:H \times H \to \mathbb{R}$ be a bounded bilinear form on Hilbert space $H$. Fix $u \in H$. Then $b(u, \cdot):H \to \mathbb{R}$ is bounded so $b(u,\cdot) \in H^*.$ Then $b(u,\cdot) = F_u(\cdot)$ ...
2
votes
0answers
68 views

An inner product on a space of linear maps

Let $V$ and $H$ be two complex Hilbert spaces. We suppose $V$ to be finite-dimensional. I'd like to understand the structure of Hilbert space on the space of linear mappings $\mathrm{Hom}(V,H)$. ...
0
votes
1answer
324 views

There exists a countable set of mutually orthogonal trigonometric functions which form a basis for $L^2(T)$. Proof?

Evidently, this fact (for real or complex valued functions) is usually taken "for-granted" in derivations of Fourier series/transform, taking $\{e^{inx}|n\in\mathbf Z\}$ as the set of basis vectors. ...
0
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1answer
23 views

A doubt in a proof concerning Hilbert spaces.

In this document about Hilbert spaces, I'm confused in the proof of Theorem 1.6 given at the bottom of pg.3. The theorem says that If $K$ is a closed convex set in a Hilbert space $H$ and $h\in ...
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0answers
219 views

Infinite dimensional vector space eigenvectors eigenvalues and representation

We can express linear transformations with their eigenvectors and eigenvalues in finite vector spaces if they are diagonalizable. even if they are not diagonalizable we can express them via Jordan ...
3
votes
2answers
62 views

If $x_n \to x$ in Hilbert space, does $|x_n| \leq C|x|$?

If $x_n \to x$ in a Hilbert space $X$, is it true that $|x_n| \leq C|x|$ for all $n$ for some constant $C$? It is true for $n$ big enough. But not sure about all $n$.
3
votes
1answer
84 views

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
1
vote
1answer
44 views

Looking for a basis of $L^2$ with this special property

The setup. Let $\mathbb{T^2}$ denote the two-dimensional torus, i.e. $$ \mathbb{T}^2 \simeq [-\pi,\pi)^2 $$ induced by identifying opposing faces of $[-\pi,\pi)^2$. Note that $$ L^2(\mathbb{T^2}) ...
1
vote
2answers
91 views

Basis means determinant of matrix of inner products is non-zero

Let $x_i$ be a basis of Hilbert space $X$ (NOT necessarily orthogonal) How do I show that $\text{det}((x_i,x_j)_H)_{ij} \neq 0$ for $i,j=1,...,n$? I see this fact used in Galerkin approximation ...
0
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1answer
126 views

A problem with linear operator in a Hilbert space

Let $(H,(\cdot,\cdot)_H)$ and $(Q,(\cdot,\cdot)_Q)$ two Hilbert separable spaces s.t $H\subset Q$ and let $B:H\to Q$ a bounded and linear operator. Let $\sigma,\tau\in H$ two fixed elements. My ...
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0answers
84 views

Continuous and dense embeddings and the density of sets in Hilbert space.

Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose ...
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vote
2answers
213 views

A problem on the bounds of Lp-norms

Let $L>0$ and $\Omega$ be the set of all integrable functions from $[0,L]$ to $[0,+\infty]$. Also, Let $f\in \Omega$ such that $\left \| f \right \|_{1}=1$. Find the tightest possible bounds for: ...
3
votes
1answer
278 views

Separable Hilbert space weak sequential compactness

To preface, Banach-Alaoglu shows weak* sequential compactness of the unit ball, and in Hilbert spaces weak* and weak convergence is the same. So I already know that the unit ball of a Hilbert space is ...
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2answers
178 views

On the uniqueness of certain weak cluster point in Hilbert space

The problem goes as follows: Let $\mathcal{H}$ be real Hilbert space, $C\subset \mathcal{H}$ be a subset. Let $\{x_n\}\subset\mathcal{H}$ satisfies the following property: ...
2
votes
2answers
111 views

Find the fallacy in using the Cauchy–Schwarz inequality

Let $\int_{a}^{b}\frac{f(x)}{x}dx=k$, wherein $f(x),a,b,k$ are positive. According to the Cauchy–Schwarz inequality: $\int_{a}^{b}xf(x)dx=\int_{a}^{b}x^{2}\frac{f(x)}{x}dx\leq \left ( ...
0
votes
2answers
41 views

Representation of a vector

$(l^2,\|\cdot\|_2)$ is a Hilbert space with scalar product $\langle x,y\rangle=\sum^{\infty}_{k=1}x_ky_k$. How can I show that every vector $x\in l^2$ can be written in a form ...
5
votes
1answer
219 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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vote
1answer
78 views

$3.\lim_{n\to\infty}\langle x,e_n\rangle=?$

given that $\{e_i\}_{n=1}^{\infty}$ is an orthonormal sequence in a hilbert space $H$, and $x\ne 0\in H$. Then could any one tell me which of the following is true? $1.\lim_{n\to\infty}\langle ...
5
votes
1answer
666 views

Prove or disprove existence of a sequence converging weakly to $0$ in an infinite dim Hilbert space

This is a problem on an old analysis qual, the prompt is: "Prove or give a counter example: if $H$ is an infinite dimensional Hilbert space and $0$ is the zero vector in $H$, then there exists a ...
4
votes
0answers
83 views

What is $H^1([0,1]) \otimes H^1([0,1])$?

Let $H^1([0,1])$ denote the Sobolev space $H^1$ on the interval $[0,1]$. What is $H^1([0,1]) \otimes H^1([0,1])$? Here, $\otimes$ the tensor product of Hilbert spaces. In particular, how is that ...
3
votes
1answer
538 views

The unit ball in a Hilbert space

I have a request for any ideas to prove: If $H$ is a Hilbert space, then any unit vector is an extreme point of the unit ball of $H$. Every isometry is an extreme point of the unit ball of the ...
4
votes
2answers
235 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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vote
2answers
73 views

A theorem about operators in Hilbert sapce

For ench $n\geq1$, $B(\mathcal{H})$ is $\ast$-isomorphic to $\mathbb{M}_n(B(\mathcal{H}))$. Thanks to the one who tell me the proof or tell me where I can find the proof.
4
votes
2answers
198 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. ...
1
vote
1answer
58 views

Hilbert space structure on Complex polynomials

Is there some sort of natural Hilbert space structure on $\mathbb{C}[z]$ so that $\{\frac{z^k}{\sqrt{k!}}\}$ are orthonormal? Can this structure be extended to $\mathbb{C}[z_1]\otimes ...
1
vote
2answers
132 views

The eigenvalues of a certain integral operator are square-summable

Let $(X,\Omega,\mu)$ be a measure space, and $k\in L^2(X\times X, \Omega\times \Omega,\mu\times\mu)$. Then it is well-known that $$(Kf)(x)=\int k(x,y)f(y)\ d\mu(y)$$ is a compact operator with norm ...
5
votes
2answers
153 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 $A$ ...
0
votes
1answer
131 views

An isometry of Hilbert spaces using the Radon-Nikodym derivative

Let $(X,\Omega)$ be a measurable space and let $\mu, \nu$ be two $\sigma$-finite measures on $(X,\Omega)$. Suppose $\nu \ll \mu$ and let $\phi$ be the Radon-Nikodym derivative of $\nu$ with respect to ...
3
votes
2answers
340 views

Net convergence and norm-convergence in Hilbert spaces

Let $\mathcal H$ be a Hilbert space which is not necessarily separable. Given a sequence of element $h_n$ indexed by $\mathbb N$, we say their sum converges in norm if the sequence $\{\sum_{n=1}^k h_n ...
1
vote
2answers
657 views

A linear manifold in a Hilbert space is dense if and only if it has trivial orthogonal complement

Conway, in A Course in Functional Analysis, leaves the following corollary (2.11) to the reader. If $\mathcal S$ is a linear manifold in $\mathcal H$, then $\mathcal S$ is dense in $\mathcal H$ ...
1
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1answer
231 views

Showing a subset of the Bergman space is closed

The following is problem 1.10 in chapter 1 of Conway's A Course in Functional Analysis. Let $G$ be an open subset of $\mathbb C$ and show that if $a\in G$, then $\{f\in L^2_a(G): f(a)=0\}$ is ...