I have a brief question regarding the norm of the $L^2$ space defined on an interval $[a,b]$. On various websites I have seen this defined as:
$\|f(x)\| = \int_{a}^{b} f(x)^2 dx$
However, yesterday I posted a question in which I had to use:
$\|f(x)\| = \sqrt{\int_{a}^{b} f(x)^2 dx}$
to get the proper answer. I asked for clarification about this in yesterday's thread, but did not receive an explanation - probably because by the time I inquired about this, my thread was already getting old. The link to my question from yesterday is:
Gram-Schmidt Orthogonalization for subspace of $L^2$
If anyone could please clarify for me exactly why we use the norm definition with a square root in the link above, I would be extremely grateful!