Prove a function is an inner product.

Question

I am supposed to prove $\langle f, g \rangle$ = $\int_{-1}^{1} f(t) g(t) dt$ is an inner product space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Our definition for inner product requires I prove 1. symmetry 2. bilinearity 3. non-negativity 4. positive definiteness. I have already proven the first two, and I think I know how to prove the last one. I'm a little confused as to how I'm going to prove $\langle f, f \rangle$ $\geq$ 0. Any help would be appreciated.

Answer 1

Non-negativity means that $\langle f,f\rangle$ is always non-negative. That's clear, since$$\langle f,f\rangle=\int_{-1}^1f^2(t)\,\mathrm dt\geqslant0.$$

Answer 2

Point $3$ and $4$ are strictly related since we need to consider

$$\langle f,f\rangle=\int_{-1}^1f^2(t)\,\mathrm dt\ge 0$$

which proves both non-negativity and positive definiteness since

$$\langle f,f\rangle=\int_{-1}^1f^2(t)\,\mathrm dt= 0 \iff f(t)=0\quad \forall t$$