I am trying to prove that $⟨f,g⟩$ is a symmetric nondegenerate bilinear form on $V$.
$V$ is simply the vector space of functions of continuous real-valued functions on the interval [0,1] ($f,g \in V$). And $⟨f,g⟩$ is defined as:
$⟨f,g⟩:= \int_{0}^1f(x)g(x)dx$
So my attempt goes as follows:
Since $f$ and $g$ are both continuous and real valued functions, then $\int_{0}^1f(x)g(x)dx$ = $\int_{0}^1g(x)f(x)dx$ and therefore $⟨f,g⟩ = ⟨g,f⟩$.
And, let $f \neq 0$, then one could define $g$ as any constant which would automatically show that $⟨f,g⟩$ is non degenerate.
Is this proof enought to show that $⟨f,g⟩$ is a symmetric nondegenerate bilinear or do I need to do a more rigorous procedure.
Thanks for the help.