I would appreciate some assistance in answering the following problems. We are moving so quickly through our advanced linear algebra material, I can't wrap my head around the key concepts. Thank you.
Let $V$ be the space of all continuously differentiable real valued functions on $[a, b]$.
(i) Define $$\langle f,g\rangle = \int_a^bf(t)g(t) \, dt + \int_a^bf'(t)g'(t) \, dt.$$ Prove that $\langle , \rangle$ is an inner product on $V$.
(ii) Define that $||f|| = \int_a^b|f(t)| \, dt + \int_a^b|f'(t)| \, dt$. Prove that this defines a norm on V.
All of the conditions for an inner product follow from properties of integrals. For example, $\int_a^b \alpha f(x)dx=\alpha \int_a^b f(x)dx, \forall\alpha \in \mathbb{R}$.