is there a method to determine one of the infinite bases of the vector space of continuous function over an interval [a,b] (a
If the question is not "well asked" : How two functions in this space are indipendent?
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First question, Axiom of choice is required, no explicit example is possible. Second question: like usual vectors: $f$, $g$ are linearly independent when $$\lambda f +\mu g=0\implies \lambda=0,\mu=0.$$