quality:correction, stylistic:style_transfer
Hilbert dimension

As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality, called the Hilbert dimension of the space. For instance, since l (B) has an orthonormal basis indexed by B, its Hilbert dimension is the cardinality of B (which may be a finite integer, or a countable or uncountable cardinal number).

The Hilbert dimension is not greater than the Hamel dimension (the usual dimension of a vector space). The two dimensions are equal if and only one of them is finite.

As a consequence of Parseval's identity, if { e } is an orthonormal basis of H, then the map Φ : H → l (B) defined by Φ(x) = ⟨x, e ⟩ is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that ⟨ Φ (x), Φ (y) ⟩ l 2 (B) = ⟨ x, y ⟩ H {\displaystyle {\bigl \langle }\Phi (x),\Phi (y){\bigr \rangle }_{l^{2}(B)}=\left\langle x,y\right\rangle _{H}} for all x, y ∈ H. The cardinal number of B is the Hilbert dimension of H. Thus every Hilbert space is isometrically isomorphic to a sequence space l (B) for some set B.