[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)

ガロア第一論文と乗数イデアル他関連資料スレ13 http://rio2016.5ch.net/test/read.cgi/math/1738367013/

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191: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2025/02/05(水) 10:50:53.01 ID:hl9U/ln8 >>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 dim

ension (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 certai

n 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 L

max 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) https://en.wikipedia.org/wiki/Hilbert_space 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 dimens

ion 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 o

f 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. http://rio2016.5ch.net/test/read.cgi/math/1738367013/191

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