# prove that a function is an inner product

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.

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You need to verify that the definitionof norm and inner products hold. Which one of the axioms are you having problems verifying ? for example, linearity for the inner product should be a simple one –  Belgi Oct 20 '12 at 16:57
I changed $<f,g>$ to $\langle f,g\rangle$. That is standard usage. –  Michael Hardy Oct 20 '12 at 17:05
You need to check (1) symmetry, ie, $\langle f,g\rangle = \langle g,f\rangle$, (2) linearity in the first argument, ie, $\langle \alpha f,g\rangle = \alpha \langle f,g\rangle$ and $\langle f_1+f_2,g\rangle = \langle f_1,g\rangle + \langle f_2,g\rangle$ and (3) positive definiteness, ie, $\langle f,f\rangle \geq 0$, with equality iff $f=0$. –  copper.hat Oct 20 '12 at 17:07
To show (ii), you need to check (1) $\|f\| \geq 0$ with equality iff $f=0$, (2) $\|\alpha f\| = |\alpha| \|f\|$, and (3) $\|f+g\| \leq \|f\| + \|g\|$. All of these properties follow from properties of the integral. –  copper.hat Oct 20 '12 at 17:20
All of the conditions for an inner product follow from properties of integrals. For example, $\int_a^b \alpha f(x)dx=\alpha \int_a^b f(x)dx, \forall\alpha \in \mathbb{R}$.
Well, positive definiteness is (slightly) trickier, for that you need to use that $f$ is differentiable. Otherwise, there'd be non-zero $f$ with $\int_a^b f^2 = 0$... –  fgp Oct 20 '12 at 17:24