# Math Help - Gram-Schmidt Process

1. ## 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.

2. 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