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Math Help - Gram-Schmidt Process

  1. #1
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    Gram-Schmidt Process

    Let H be a seperable Hilbert space and M a countable dense subset of H.

    Show that H contains a total orthonormal sequence which can be obtained from M by the Gram-Schmidt process.
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  2. #2
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    Let B= \{e_k : k \in \mathbb{N} \} be your countabl dense subset of H, and B_1= \{w_k : k \in \mathbb{N} \} the sequence obtained by applying the Gram-Schmidt process to e_k. Then let w_i and w_j be two elements in B_1 and m=max \{i,j \} then applying G-S to the first m vectors we conclude that w_i \perp w_j. So B_1 is an orthonormal set.

    Now, without loss of generality, we may assume that B is linearly independent, and since B_{1,m} = \{w_k : k \leq m \} is contained in span(B) for every m \in \mathbb{N} we have B_1 \subset span(B). Let H_1 = span(B), since B_1 is linearly independent (actually orthogonal) and \vert B_1 \vert = \vert B \vert we have (remember that all basis for a vector space have the same cardinality) that B_1 is a basis for H_1 and so span(B_1) = H_1. And you're finished
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