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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 $$\lambda f +\mu g=0\implies \lambda=0,\mu=0.$$

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  • $\begingroup$ But saying when two vectors are independent is not enough to define linear independence. $\endgroup$ – Marc van Leeuwen Oct 2 '14 at 17:44
  • $\begingroup$ @MarcvanLeeuwen, true, but the OP question was "How two functions in this space are indipendent?" $\endgroup$ – Martín-Blas Pérez Pinilla Oct 3 '14 at 6:13

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