I'm trying to prove that the space of real-valued functions on the closed interval $[a,b]$ where $a < b$ is an infinite dimensional vector space over $\mathbb{R}$.
$V$: The space of real-valued functions on the closed interval.
$W$: Subset of $V$ where all the coefficients are set to 1.
If I consider the polynomials $1,x,x^2,~…~x^n$ (Where all these polynomials are linearly independent), I can repeatedly create a polynomial $x^{n+1}$ which is linearly independent, and thus the basis is of infinite order for $W$, $W$ is infinite dimensional and this implies that $V$ is infinite dimensional.
Am I getting anywhere? Would appreciate feedback.
Best Answer
Yes, you are getting somewhere. But you are not expressing yourself in the best possible way. First of all what are the coefficients of a function? In the second place, you should define $W$ as the space of the polynomal functions from $[a,b]$ into $\mathbb R$. Yes, $W\subset V$ and, yes, $W$ is infinite-dimensional (from which you can deduce that $V$ itself is infinite-dimensional). The reason for this is that, for each $n\in\mathbb N$, $x^n\notin\langle1,x,x^2,\ldots,x^{n-1}\rangle$.