Here's my attempt:
$$L^2$$ is closed under addition. I need to show that $$ \left(\int_a^b |f(t)|^2 \, dt\right)^{\frac{1}{2}}<\infty$$ and $$\left(\int_a^b |g(t)|^2 \, dt \right)^{\frac{1}{2}}<\infty$$ then $$\left(\int_a^b |f(t)+g(t)|^2 \, dt\right)^{\frac{1}{2}}<\infty$$
Invoking the Cauchy-schwartz inequality, we have: $$\left \| f+g \right \|_{2}^{2} = \left \langle f+g,f+g \right \rangle = \left \langle f,f \right \rangle+\left \langle g,g \right \rangle +\left \langle f,g \right \rangle+\left \langle g,f \right \rangle = \left \| f \right \|_{2}^{2}+\left \| g \right \|_{2}^{2}+2Re\left \langle f,g \right \rangle \leq \left \| f \right \|_{2}^{2} + \left \| g \right \|_{2}^{2}+2\left \| f \right \|_{2}\left \| g \right \|_{2}$$
Could someone give me a little push?
Note $$\|f\|_2^2 + \|g\|_2^2 + 2\|f\|_2 \|g\|_2 = (\|f\|_2 + \|g\|_2)^2,$$
so you have $\|f + g\|_2^2 \le (\|f\|_2 + \|g\|_2)^2$. Now take square roots.