This is my pde homework
$3$. Consider the family of functions $\beta_n=\sin(2\pi nx/L)$.
$\qquad(a)$ Show that this family is orthogonal, but not complete, within the space of piecewise $\:\quad\qquad$continuous functions on the interval $[0,L]$.
$\qquad(b)$ Is it complete for the interval $[0,L/2]$?
Here is my attempt, I have proved that this family is orthogonal, but I do not understand what this question wants me to prove next.
$\langle\beta_n,\beta_m\rangle=\int_0^L\sin(n\pi x/L)sin(m\pi x/L)dx=0$
$\langle\beta_n,\beta_n\rangle=\int_0^L\sin^2(n\pi x/L)dx={L\over 2}$
Googling "complete orthogonal" leads to this link.
To paraphrase, an orthogonal system ($\beta_n$ in your case) is called complete if every piecewise continuous function can be approximated in the $L^2$-norm by (finite) linear combinations of members of the orthogonal system.
Note that definitions such as these are sometimes used differently by different people and it is best to check the precise definition in your course material.