[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ13 (1002ﾚｽ)
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191: 現代数学の系譜 雑談 ◆yH25M02vWFhP 02/05(水)10:50 ID:hl9U/ln8(1/5) AAS
>>182 補足

・Hilbert spaceの Hilbert dimension は、下記
"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.[94]"
(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).”
　”As a consequence of Parseval's identity,[95] 略 ”
・なお、>>146-147 "Proof that every vector space has a basis"では、有限和は 陽には使われていない
　なので ”The set X is nonempty since the empty set is an independent subset of V, and it is partially ordered by inclusion, which is denoted, as usual, by ⊆.
　Let Y be a subset of X that is totally ordered by ⊆, and let LY be the union of all the elements of Y (which are themselves certain subsets of V).
　Since (Y, ⊆) is totally ordered, every finite subset of LY is a subset of an element of Y, which is a linearly independent subset of V, and hence LY is linearly independent. Thus LY is an element of X. Therefore, LY is an upper bound for Y in (X, ⊆): it is an element of X, that contains every element of Y.
　As X is nonempty, and every totally ordered subset of (X, ⊆) has an upper bound in X, Zorn's lemma asserts that X has a maximal element. In other words, there exists some element Lmax of X satisfying the condition that whenever Lmax ⊆ L for some element L of X, then L = Lmax.”
　とやっているので、⊆ による順序は Hilbert space でも そのまま使える
　あとは、直交基底と 位相的な収束の話を 色付けすれば、よさそうだ

(参考)
外部ﾘﾝｸ:en.wikipedia.org
Hilbert space

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.[94] For instance, since l^2(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).

As a consequence of Parseval's identity,[95] if {ek}k ∈ B is an orthonormal basis of H, then the map Φ : H → l^2(B) defined by Φ(x) = ⟨x, ek⟩k∈B 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
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^2(B) for some set B.

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