## New answers tagged hilbert-spaces

2
votes

Accepted

### Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II

Gro-Tsen's answer to your previous question provides a counterexample if you define $D$ to be all vectors in $\ell_2$ that are of the form $\sum_n a_n f_n$, where
$f_n = e_n + e_{n+1}$, $(e_n)$ is the ...

- 30k

6
votes

### Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

I believe the following is a simple counterexample:
Let $(X,\mu)$ be $\mathbb{N}$ with the counting measure (so I will be writing $\ell^2$ for $L^2(X,\mu)$. Let $g=0$ and $\tilde g(n) = (-1)^n$. Let ...

- 23.7k

6
votes

Accepted

### Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

The answer is no and the following result provides a quite interesting counterexample. This is a known result, but I am not sure where to find it in the literature.
Theorem. If $f\in L^1_{\rm loc}(\...

- 23.8k

4
votes

Accepted

### Left and right eigenvectors are not orthogonal

Yes, this is always true if $\lambda \not= 0$. The subsequent theorem shows a more general result. To formulate it, we need the following terminology:
For an eigenvector $\lambda$ of a bounded linear ...

- 9,846

2
votes

### Motivation for Heisenberg's modeling of observables

Sorry, for self answer, but I think this is what's happening. I don't know how this is related to Connes' explanation though.
Any measurement can be interpreted as a combination of 'yes-no' ...

- 273

0
votes

### Function monotony between [0,T] and $L^2$

First, since you have $H^1(0,T)$ imbedds in $\mathscr{C}^0([0,T])$, $z$ can be seen as an element of $\mathscr{C}^0([0,T];L^2(\Omega))$ and you can speak without ambiguity of $z(t_1)$ and $z(t_2)$. ...

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