What is the range of the transformation $T: C[0, 1] \to \mathbb R$ defined by $T(f) = \int_0^1 f(x)\,dx$
$C[0, 1]$ is the vector space of all continuous functions $f: [0,1] \to \mathbb R$.
So the range of this transformation is all $a \in \mathbb R$ such that $T(f) = a$, correct? So would that make the range all of $\mathbb R$, or am I doing something wrong?
I'm generally confused about the idea of range when you are mapping from a vector space to something that isn't a vector space. If someone could touch on this topic it would be great :)
$\mathbb R^1 = \mathbb R$ is in fact a vector space of one dimension. Check the vector space axioms if you're still unsure.
For all $c \in \mathbb R$, consider the constant function $f(x) = c$. Clearly $f \in C[0, 1]$. Furthermore, $T(f) = \int_0^1 c \,dx = c$. Therefore, the range of $T$ is all of $\mathbb R$.
Another way to approach it, since $T$ is linear and $\mathbb R$ is one dimensional, the range of $T$ must be zero or one-dimensional. Since we know that the range isn't only $\{0\}$, the range must be one-dimensional, hence all of $\mathbb R$.