I was just introduced to Hilbert Spaces, and am wondering what one of the formulas I'm using is called.

Question

This is the formula/normalization? Is there any specific name for this? I'm not sure what to search up to find information about it.

$\langle f | f\rangle = \int_a^b|f(x)|^2 d x$

Answer 1

If $\langle \cdot | \cdot \rangle$ is the inner product of a Hilbert space, then $f \mapsto \| f \| := \sqrt{\langle f | f \rangle}$ induces a norm. In your case, it is the norm on $L^2(a,b)$.