How to construct an arbitrary set of linearly independent linear maps on $C({\infty})$ functions on R?
My attempt: I can choose S = {1, 2, ..., k} contained in R where k is arbitrary. For each i in S, I can create a bump function $f_i$ in a $\epsilon$ nbd of i. These functions will be linearly independent. Since k is arbitrary, $C^{\infty}(R)$ is infinite dimensional.
Now, I define $T_i(f)=\int_{i-\epsilon}^{i+\epsilon}{f_i(x)f(x)dx}$. Then $T_i$'s are linear and linearly independent. Just wanted to know if this approach is right?