When a set of functions becomes complete? I know that a set of functions are said to form a complete basis on an inteval if any function on that interval can be expressed as a linear combination of the functions in the set. I also know that every function in the set are  orthogonal.
Now what is what is the condition(s) that a set of functions has to statisfy to become complete? 
That is, how to prove a set of orthogonal functions span a space? 

A metric space is said to be "complete" if every Cauchy sequence converges.
For example: Let $(X, \mu)$ be a measure space. Then $L^P(X)$ is complete under the $L^P$ norm, for $p \in [1,\infty]$. [It is a Banach space.]
Every finite dimensional normed vector space is also complete. (This this can be explained by the Lipschitz equivalence to the euclidian norm.)
Notions of completeness need not be restricted to a set of functions. For example, $\mathbb{R}$ is complete, since every Cauchy sequence $\{a_n\}$ converges in $\mathbb{R}$.
In fact, it can be proven that $\mathbb{R}$ is  the completion of $\mathbb{Q}$; i.e.: take a sequence of rationals that is Cauchy and define it's limit to be a real number.
