L2 and Scalar Product

Question

I struggle to show that for $\, f,g\in L^2(X,A,\lambda)$ this is a scalar product: $ \langle f,g\rangle := \int fg \, d\lambda $

It wasn't hard to show that the length is positive, but I struggle with the symmetry and the linearity.

Answer 1

Hint:

Symmetry is trivial. $\left<f,g\right> = \int fg d\lambda = \int gf d\lambda=\left<g,f\right>$

because $fg(x)=f(x)g(x)=g(x)f(x)=gf(x)$.

For linearity you need to use the fact that

$$\int (f+g) d\lambda = \int f d\lambda + \int g d\lambda$$