a question on completeness within functional analysis

suppose I have a function f(x,t) where both x and t are continuous, I treat x as variable, but t as parameter, as t varies, I get a sequence of f(x), I want to know whether this sequence of f(x) is complete to build up all possible functions on x.

This should be a well-defined problem, but maybe the tones are unclear. First, should I introduce say L2 norm when talking about completeness? Second, can anybody lead me to look into this problem?

Thanks,