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

if \(M\) is total in \(X\), then \(M^\perp = \{0\}\).

if \(X\) is complete and \(M^\perp = \{0\}\) then \(M\) is total in \(X\).

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 \(x \in X\).

Proof:

Will be added later.

Lemma 1: in every non-empty Hilbert space there exists a total orthonormal set.

Proof:

Will be added later.

Theorem 2: all total orthonormal sets in a Hilbert space have the same cardinality.

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

if \(X\) is separable, every orthonormal set in \(X\) is countable.

if \(X\) contains a countable total orthonormal set, then \(X\) is separable.

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.

Proof:

Will be added later.