Hi I am trying to understand a claim made in my textbook and was wondering if anyone can show formally that the set of functions $g_n$ is indeed an orthogonal set.
The claim:
Let $f_1, f_2, f_3, \dots$ be an independent set of functions in $L^2$. Define the set of functions $g_n$ by
$g_1 = f_1$,
$g_2 = f_2 - \frac{(f_2, g_1)}{||g_1||^2}g_1$,
$g_3 = f_3 - \frac{(f_3, g_2)}{||g_2||^2}g_2 - \frac{(f_3, g_1)}{||g_1||^2}g_1$,
What would be the way to show mathematically that $g_n$ is an orthogonal set?
Proceed by induction.
First show that $g_2 \bot g_1$ by evaluating $\langle g_{2}, g_1 \rangle$ and showing that it is zero.
Now suppose that the $g_1,...,g_n$ are orthogonal.
Now show that $g_{n+1} \bot g_k$ for $k=1,...,n$ by computing $\langle g_{n+1}, g_k \rangle$ and showing that it is zero.
In order to show that the $g_k$ are non zero, you need to use independence of the $f_k$.
Again, we use induction. We see that $\operatorname{sp} \{ g_1 \} = \operatorname{sp} \{ f_1 \}$.
Now suppose $\operatorname{sp} \{ g_1,...,g_n \} = \operatorname{sp} \{ f_1 , ..., f_n\}$. Then the formula for $g_{n+1}$ shows that if $g_{n+1} = 0$, we would have $f_{n+1} \in \operatorname{sp} \{ f_1 , ..., f_n\}$, which would contradict the $f_k$ being independent.
Since $g_{n+1} \in \operatorname{sp} \{ g_1,...,g_n, f_{n+1} \} $, we see that $g_{n+1} = \alpha f_{n+1} + \sum_k \alpha_k g_k$ and $\alpha \neq 0$, hence $f_{n+1} = {1 \over \alpha} ( g_{n+1} - \sum_k \alpha_k g_k )$, so we see that $\operatorname{sp} \{ g_1,...,g_n, f_{n+1} \} = \operatorname{sp} \{ g_1,...,g_n, g_{n+1} \} = \operatorname{sp} \{ f_1 , ..., f_n, f_{n+1}\}$.