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Linear subspace/Dual space/Orthogonal space/Correspondence/Fact/Proof

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Proof

(1) and (2) are clear. (3). The inclusion

is also clear. Let , . Then we can choose a basis of and extend it to a basis of . The linear form vanishes on , therefore, it belongs to . Because of

we have .

(4). Let be a basis of , and let

denote the mapping where these linear forms are the components. Here, we have

Assume that the mapping is not surjective. Then is a strict linear subspace of and its dimension is at most . Let be a -dimensional linear subspace with

Due to fact, there is a linear form

whose kernel is exactly . Write . Then

contradicting the linear independence of the . Moreover, is surjective and the statement follows from fact.