I have a homework problem that I can't figure out...
Is the space of all real valued continuous functions defined on the interval [0,1] finite dimensional?
please help me! thanks.
You need to be more formal. If $\displaystyle \mathcal{C}[0,1]$ (continous functions) were to be a finite dimensionsal vector space then every subspace would be finite dimensional also, with a smaller degree (this is a theorem). But then $\displaystyle \mathcal{P}[0,1]$ (polynomials functions) would be finite dimensional. But that is impossible. Why?
Again, why is P[0,1] infinite? You need to be more formal. Here is how you show it cannot have a finite basis. Suppose that P[0,1] has a finite basis $\displaystyle \{ p_1(x),...,p_n(x)\}$ where $\displaystyle p_i(x)$ are polynomials. Then it would mean any polynomial in P[0,1] can be expressed in the form $\displaystyle a_1p_1(x)+...+a_np_n(x)$. Let $\displaystyle f(x)$ be a polynomial with a larger degree then all of the $\displaystyle p_i(x)$ polynomials. Then $\displaystyle f(x)=a_1p_1(x)+...+a_np_n(x)$ is impossible because polynomials can only be equal if their have the same coefficients and the same degree, but it is impossible since the degree of f(x) exceedes the RHS.