I'm reading Applied Partial Differential Equations by DuChateu and Zachmann, and the first couple of chapters contain quite a bit of review of Fourier series, as well as theory about L2 integrable functions and orthogonal/orthonormal basis sets of functions.
Some of the exercises require showing that a particular family of functions is not a complete orthogonal or orthonormal family over a certain interval. The definition given by the book which I am able to remember is that a family of functions is not complete in L2 if $\exists$ a non-zero function $g \in L^2: (g, u_k) = 0$, $\{u_k: k = 1, 2,\ldots\}$ The full definition of a complete family of functions is here.
There don't appear to be any exercises on how to prove a family of functions is complete; it would seem this is a more difficult task. For example, the family of functions $(\frac{2}{\pi})^{\frac{1}{2}}\sin(kx)$ is stated to be complete and orthonormal on the interval $L^2(0, \pi)$; IIRC the family of function $\sin(\frac{n\pi x}{L})$ is complete and orthogonal (but not orthonormal) on the interval $(0, L)$.
I don't think the definition in 2 would be much help, certainly one can't evaluate every possible piecewise continuous function on the interval in question? The (perhaps simpleminded) approach I had in mind (assuming the converse of the the above definition of a function not being complete is true) would be to go from there with a proof by contradiction, or else somehow show that if there is some function g that purports to make the inner product $(g, u_k)$ zero on the interval, that this function by necessity would have to converge to $u_j, j \neq k$ at all points on the interval to make the inner product integral go to zero. I haven't gotten very far in my attempts, though, and any further advice on how to go about this would be appreciated.