How to find the dimension of a function? I just get confused of that:
Consider the set of continuous functions on the interval $[0,1]$:
$$C[0,1]=\{f:[0,1]\mapsto \mathbb{R} | f\text{ is continuous}\}$$
Then, what is the dimension of $C[0,1]$?
How can I find the dimension of $C[0,1]$ and how can I verify that?
Best Answer
$C[0,1]$ is an infinite-dimensional space. For example, $1, x, x^2, x^3 , \ldots$ are linearly independent.