Tagged Questions

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

0answers
68 views

Prove that if $X$ is a Hilbert space, then $B(X)$ is not a Hilbert space

Im having a homework question that goes like this: X is a Hilbert space, a complete inner product space, show that B(X) is not a Hilbert space. Im quite stuck and I would love to understand this ...
1answer
133 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
1answer
75 views

I dont understand this notation

I`m having a homework question that goes like this: $X$ is a Hilbert space, a complete inner product space, show that $B(X)$ is not a Hilbert space. My only question for now is what does $B(X)$ means?...
0answers
48 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
1answer
2k views

Operator norm of orthogonal projection

I was assigned the following homework problem: "Let $P:\mathcal{H} \to \mathcal{H}$ be bounded and linear. Assume it satisfies $P^2 = P$ and $P^\star = P$. Show $\|P\| \le 1$." This isn't too hard ...
0answers
53 views

The subspace sum of closed subspaces is closed [duplicate]

Given an arbitrary Hilbert space $\scr H$ and closed subspaces $A,B\subseteq\scr H$ with trivial intersection, is it true that $A+B=\{x+y:x\in A,y\in B\}$ is closed? So far, I have the following: Let ...
1answer
147 views

2answers
49 views

Orthogonal Projector

Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$. $P_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. I have to prove that $P_{\psi}$ is an ...
2answers
118 views

Unitary operator on dense set, Unique extension?

given two Hilberspace $H_1$ and $H_2$. Let $V\subset H_1$ and $W\subset H_2$ be dense subspaces. Furthermore let $U: V \rightarrow W$ be an unitary operator. I just want to know whether there is a ...
0answers
37 views

Limit of function of an operator

Let $A_n$ be a sequence of bounded, self-adjoint operators on Hilbert space $\mathcal{H}$. Let us assume that for some vector $\psi\in\mathcal{H}$, $$\lim_{n\rightarrow\infty}A_n\psi = \alpha \psi.$$...
1answer
66 views

isometric embedding of l^2

CLAIM: Let $H$ be an infinite dimensional $\mathbb{R}$-Hilbert space. Then the $\ell^2$ sequence space can be embedded in $H$. I think it could be true since every Hilbert space has an orthonormal ...
2answers
30 views

1answer
102 views

Weak convergence of subsequence in Hilbert spaces

Prove that if $x_n$ is a sequence in $H$ (Hilbert space) with $\sup_n||x_n||\le1$, then there is a subsequence $\{x_{n_j}\}$ and an element $x$ of $H$ with $||x||\le 1$ such that $x_{n_j}$ converges ...
2answers
70 views

2answers
113 views

showing uniqueness of a Hahn Banach extension

I am trying to prove the following: If $H$ is a Hilbert space and $G\subseteq H$ is a closed linear subspace, then any bounded linear functional on $G$ has a unique Hahn-Banach extension on $H$. So ...
1answer
140 views

1answer
69 views

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$.

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$. Let consider $H=l_2$ where $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace$ ...