Prove that $\|f\|_2 :=\left( \int_0^1 |f(t)|^2 dt \right)^{1/2}$ defines a norm

This question is an exercise from the first chapter of Topology of metric spaces by S.Kumaresan

Let $X := [0,1]$, the closed unit interval. Then$$\|f\|_2 := \left(\int_0^1 |f(t)|^2 dt\right)^{1/2}$$defines a norm on the set of all continuous real/complex valued functions on $[0,1]$.

I was able to prove this for real function, I used the fact $|f(x)|^2 = (f(x))^2$ along with Cauchy-Schwarz inequality as in case of vector space $C^1([0,1])$$$\langle f,g\rangle = \int_0^1 f(x)g(x) dx$$ is a inner product.

But I'm not sure how to proceed with the complex function case, as I have not studied complex analysis yet.