Question about inner product of $C_{2}[a,b]$.

Question

My question, I think, is quite simple. I have a space $C_{2}[a,b]$ of complex-valued continuous functions. I checked that functional $$(f,g) = \int_{a}^{b} f(t)\overline{g}(t)dt$$ is a inner product on this space. But I don't know how should I work with them. For example, I have a function $g(t) = t^3$ hence $g(t) = (x+iy)^3$ and $\overline{g}(t) = (x-iy)^3$? I should consider any function $u(x,y)$ as sum: $u(x,y) = f(x,y) + ih(x,y)$? Thank you very much and sorry for pretty elementary question..

Answer 1

A function $f:[a,b]\to\mathbb C$ is of the form $f(t)=f_1(t)+if_2(t)$ where $f_1$ and $f_2$ are real valued functions.

In particular, for $g(t)=t^3$, this is just a real valued function.