Pythagoras theorem for infinite sets of vectors

Question: Is it possible to get an analogue of the Pythagoras theorem for infinite sets of pairwise orthogonal vectors?

Answer: Yes. We are not going to give the most general form of the Pythagoras theorem. It would lead us to Hilbert and Banach spaces, too far from the Linear Algebra course. Consider an example.

Take the Euclidean vector space C[-Pi,Pi] of all continuous functions on the interval [-Pi,Pi]. Consider the following set of functions: sin(x), cos(x), sin(2x), cos(2x),..., sin(nx), cos(nx), ... . One can prove that these functions are pairwise orthogonal, that is

int(sin(mx)sin(nx), x=-Pi..Pi) dx = 0 if n is not equal to m;
int(sin(nx)cos(mx), x=-Pi..Pi) dx = 0 for every m and n.

and the norm of each of them is sqrt(Pi). Let f(x) be any function from C[-Pi;, Pi]. Then from the second semester of calculus we know that f is representable by the Fourier series:

f(x)= sum (a_n sin(nx)+ b_n cos(nx))

Then it is possible to prove that

||f(x)||²=sum(||a_n sin(nx)||²+||b_n cos(nx)||²) = π*sum(a_n²+b_n²).

This is called the Parseval equality (or the Energy equation). This result can be generalized to other infinite sets of pairwise orthogonal functions.

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