$H=\{f\in C[0,1]:f(0)=0, f \text{ is absolutely continuous, and } f' \in L^2[0,1]\}$
(a). Prove that $\langle f,g\rangle =\int_0^1 f'g'dx $ defines an inner product on $H$.
(b). Prove that the injection $i : H \to C[0,1]$ is continuous.
For the first part, I know to show that it defines an inner product, I show that for $f,g,h \in H$ and $\alpha, \beta \in R$, $$(i). \langle f,f\rangle \geq 0,\text{ and }\langle f,f\rangle=0 \iff f=0 $$ $$(ii). \langle \alpha f+\beta g ,h\rangle =\alpha\langle f,h\rangle +\beta\langle g,h\rangle $$ $$(iii). \langle f,g\rangle =\langle g,f\rangle $$ $(ii)$ and $(iii)$ seem to be easy to show. But regarding $(i)$ this is what I have $$\langle f,f\rangle =\int_0^1 (f')^2dx \geq 0$$ since $(f')^2 \geq 0$.
Assuming $\langle f,f\rangle=0$, I need to show that $f=0$. $$\langle f,f\rangle =\int_0^2 (f')^2=0 \Rightarrow (f')^2=0 \Rightarrow f'=0.$$ My concern is that $f'=0$ does not necessarily imply $f=0$.
Does the fact that $f(0)=0, f\in H$ do any help here?
Regarding problem (b), I don't really know how to go by that, can someone provide any help?