59 lines
1.6 KiB
Markdown
59 lines
1.6 KiB
Markdown
|
# Total sets
|
||
|
|
|
||
|
|
> *Definition 1*: a **total set** in a normed space $(X, \langle \cdot, \cdot \rangle)$ is a subset $M \subset X$ whose span is dense in $X$.
|
||
|
|
|
||
|
|
Accordingly, an orthonormal set in $X$ which is total in $X$ is called a total orthonormal set in $X$.
|
||
|
|
|
||
|
|
> *Proposition 1*: let $M \subset X$ be a subset of an inner product space $(X, \langle \cdot, \cdot \rangle)$, then
|
||
|
|
>
|
||
|
|
> 1. if $M$ is total in $X$, then $M^\perp = \{0\}$.
|
||
|
|
> 2. if $X$ is complete and $M^\perp = \{0\}$ then $M$ is total in $X$.
|
||
|
|
|
||
|
|
??? note "*Proof*:"
|
||
|
|
|
||
|
|
Will be added later.
|
||
|
|
|
||
|
|
## Total orthornormal sets
|
||
|
|
|
||
|
|
> *Theorem 1*: an orthonormal sequence $(e_n)_{n \in \mathbb{N}}$ in a Hilbert space $(X, \langle \cdot, \cdot \rangle)$ is total in $X$ if and only if
|
||
|
|
>
|
||
|
|
> $$
|
||
|
|
> \sum_{n=1}^\infty |\langle x, e_n \rangle|^2 = \|x\|^2,
|
||
|
|
> $$
|
||
|
|
>
|
||
|
|
> for all $x \in X$.
|
||
|
|
|
||
|
|
??? note "*Proof*:"
|
||
|
|
|
||
|
|
Will be added later.
|
||
|
|
|
||
|
|
> *Lemma 1*: in every non-empty Hilbert space there exists a total orthonormal set.
|
||
|
|
|
||
|
|
??? note "*Proof*:"
|
||
|
|
|
||
|
|
Will be added later.
|
||
|
|
|
||
|
|
> *Theorem 2*: all total orthonormal sets in a Hilbert space have the same cardinality.
|
||
|
|
|
||
|
|
??? note "*Proof*:"
|
||
|
|
|
||
|
|
Will be added later.
|
||
|
|
|
||
|
|
This cardinality is called the Hilbert dimension or the orthogonal dimension of the Hilbert space.
|
||
|
|
|
||
|
|
> *Theorem 3*: let $X$ be a Hilbert space, then
|
||
|
|
>
|
||
|
|
> 1. if $X$ is separable, every orthonormal set in $X$ is countable.
|
||
|
|
> 2. if $X$ contains a countable total orthonormal set, then $X$ is separable.
|
||
|
|
|
||
|
|
??? note "*Proof*:"
|
||
|
|
|
||
|
|
Will be added later.
|
||
|
|
|
||
|
|
> *Theorem 4*: two Hilbert spaces $X$ and $\tilde X$ over the same field are isomorphic if and only if they have the same Hilbert dimension.
|
||
|
|
|
||
|
|
??? note "*Proof*:"
|
||
|
|
|
||
|
|
Will be added later.
|
||
|
|
|