- 710

- 5

if we consider the vector spaces of integrable real functions on

*[a,b]*with the inner product defined as: [tex]\left \langle f,g \right \rangle=\int _a^bf(x)g(x)dx[/tex] the Cauchy-Schwarz inequality can be written as: [tex]\left | \int_{a}^{b} f(x)g(x)dx\right | \leq \sqrt{\int_{a}^{b}f(x)^ 2dx} \sqrt{\int_{a}^{b}g(x)^ 2dx}[/tex]

Does it still hold true that, like in ℝ

^{n}, equality holds

*iff*[itex]g=\lambda f[/itex] for some real scalar λ?