Reconciling different definitions of orthogonality I want to establish about orthogonality in my mind. I knew the orthogonality of two functions $f$ and $g$ in interval $T$ like the following: $$ \int_{\langle T\rangle}f(t)g^*(t)~dt=0 \tag{1} $$ where $$ g^*(t) \mbox{ is a conjugate of } g(t) $$ However, in some textbooks, if $$ \text{E}\left[f(t)g(t)\right]=0 \tag{2}\ , $$ then two functions $f$ and $g$ are orthogonal. I want to know the relationship between two equations.

The notation $E(f)$ is most commonly used in probability theory and this means simply just $\int_{X}fd\mu$ where $(X,\mu)$ is your measure space (it is called _expectation_ ). Therefore the only difference between those two relations is the fact that the second uses _real_ scalar product. In other words $g$ is real if and only if $\overline{g(t)}=g(t)$ for all $t$ and in this situation these two notions coincide.