notes/docs/mathematics/functional-analysis/inner-product-spaces/total-sets.md

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.

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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.

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

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

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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.

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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.