Let us denote by $U$ the following subspace of $C^{1}[a, b]$ :
$$
U=\left\{f \in C^{1}[a, b]: f(a)=0\right\}
$$
and with $W$ the following subspace of $C^{1}[a, b]:$
$$
W=\left\{f \in C^{1}[a, b]: f=\text{const on }[a, b]\right\}.
$$
I want to show the following isomorphisms of vector spaces:
$ (a) \quad C[a, b]\cong U\\
(b) \quad C[a, b] \cong C^{1}[a, b] / W.
$
Any hints are appreciated. Thanks in advance.
Dimensions are not useful at all. For a) consider the map $f \mapsto \int_a^{x}f(t)dt$. Verify that this map is linear, one-to-one and onto.
For b) consider the equivalence class of a function $f\in C^{1}([a,b])/W$. Map this to the derivative $f'$. Check that this is linear , one-to-one and onto. [Recall that $f'=g'$ implies that $f-g \in W$].