Consider $L(f)=\int_{a}^{b}f(x) dx$ for all $ f∈C[a,b]$
$L$ maps $C[a,b] \to \mathbb{R^1}$
Why does performing integration on a function output something that is in $\mathbb{R^1}$?
If that was not worded correctly, please correct me. I can clarify as well.
It's a definite integral. It maps the function to the signed area under $y = f(x)$, between $x = a$ and $x = b$, which is a real number.