Show that the vector space $C(\Bbb R)$ of all continuous functions defined on the real line is infinite-dimensional.
I get that if $C(\Bbb R)$ contains an infinite-dimensional subspace, then it is infinite-dimensional, but how do I prove that? Obviously $\Bbb R$ is infinite…
Best Answer
Consider the subspace of $C(\Bbb R)$ whose vectors are the polynomials. This subspace has the following basis: $$\langle1,x,x^2,x^3,x^4,\dots\rangle$$ Each element is linearly independent of all the others. To show this, suppose there exists a linear combination that evaluates to zero everywhere: $$\sum_ia_ix^i=0$$ The left-hand side is a polynomial with an infinite number of roots. Any non-zero polynomial, however, must have a finite number of roots (no more than its degree). Therefore the left-hand side is the zero polynomial, i.e. $a_i=0$ for all indices $i$.
Since there are an infinite number of elements in the basis, the subspace and thus $C(\Bbb R)$ are infinite-dimensional.