Cauchy–Schwarz inequality for complex-valued functions

Question

Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{C}$. I wonder if the following inequality, with no absolute value, is true: $$\left(\int_0^1 f(x) \overline{g(x)}\ dx\right)^2\leq \int_0^1 f(x)^2\ dx\int_0^1 g(x)^2\ dx$$

Answer 1

That is not true.

Consider $f(x)=g(x)=\frac{1+i}{\sqrt{2}}$.

Observe that

$$ \int_{0}^{1} f(x)\overline{g(x)}dx = \int_{0}^{1} 1 dx =1$$

And note that

$$\int_{0}^{1} f(x)^2dx=i =\int_{0}^{1}g(x)^2 dx $$

Thus, if the inequality is true,

$$(1)^2\leq (i)(i)=-1 $$

And it is a contradiction.

Thus, the given f(x) and g(x) make a counter example of your inequality.