Talk:Dual space/Archive 2
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Quotient spaces and annihilators: "equality holds provided V is finite-dimensional"
The article currently says that A^0 + B^0 = (A \cap B)^0 and equality holds when V is finite-dimensional. This is always true, and the only reason to assume finite-dimensionality is to make this easier to prove. If we are not providing a proof anyway, then I don't see any reason to assume that V is finite-dimensional. 50.1.105.93 (talk) 06:24, 7 July 2016 (UTC)
Naturally induced linear structure
"In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V together with a naturally induced linear structure." Here, "naturally induced linear structure" is meaningless, so this sentence does not serve as a definition as the following sentence claims. 96.255.35.216 (talk) 13:11, 19 August 2017 (UTC)
Meaning of natural pairing not clear
In the second paragraph under "Algebraic dual space", it says, "The pairing of a functional in the dual space and an element of ... is called the natural pairing." Yet it doesn't explain what this natural pairing is, or under what conditions it exists. The linked article on a dual pair seems to suggest the word "pair" refers to the spaces themselves, not the vectors within. Is there some natural way to match vectors when the vector spaces form a dual pair?
Tnedde (talk) 15:35, 16 June 2018 (UTC)
- Since the bilinear mapping defined by ⟨·,·⟩ : V∗ × V → F maps all ordered pairs from the cross product, V∗ × V, it is clear that any functional φ in the dual space can be paired with any element x of V. Thus all such pairs are natural pairings.—Anita5192 (talk) 03:38, 17 June 2018 (UTC)
Brackets
In the very first section after the table of contents, it's suggested that arguments in square brackets be ordered thus [x,φ] and in angled brackets thus <φ,x>. But by the time we get to 'Transpose of a linear map', you are using square brackets with the second order. 37.205.58.146 (talk) 12:58, 1 May 2019 (UTC)