Dimension of vector space of real-valued functions over $R$ 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. 

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 $n\in\mathbb N$, $x^n\notin\langle1,x,x^2,\ldots,x^{n-1}\rangle$.
