"Scalar product" of two Lp spaces

Question

I was reading the book A. Lasotta and M. C. Mackey, "Chaos, Fractals, and Noise: Stochastic Aspects of Dynamic", Springer, 1991

On page 27, they defined a ``scalar product'' as follows. Let $f\in\mathcal{L}^p(X)$ and $g\in\mathcal{L}^q(X)$, where $0<p<q\leq\infty$ and $p^{-1} + q^{-1}=1$. A scalar product can be defined as follows

$$\langle f,g\rangle=\int_X fg$$

I would like to know

1) On which space is the scalar product defined: $\mathcal{L}^p(X)$ or $\mathcal{L}^q(X)$?

2) It is indeed a scalar product?

I have a counter example for item 2: let $X=[0,1]$, $p=1$, $f(\cdot)=1/(\cdot)^{1/2}$. Thus, $q=\infty$ and it cannot be an scalar product, since $$\langle f,f\rangle=\int_0^1\frac{1}{x}\,dx=\infty$$

Does someone knows how how to answer 1 or if my counterexample for 2 holds true?

Ps.: $\mathcal{L}^p(X)$ is the class of $p$-Lebesgue integrable functions with the norm

$$|f|_p=\left(\int_X|f|^p\right)^{1/p}<\infty$$

Answer 1

"Scalar product" is a misnomer. It's actually a duality, that is, a bilinear map $L^p\times L^q\to\mathbb F$.

Answer 2

The counterexample doesn't hold true: Because $\sup_{x\in[0,1]}x^{-\frac{1}{2}}=\infty$ it follows $f\notin L^{\infty}([0,1])$, so the pairing $\langle f,f\rangle$ is not defined.For $f\in L^p,g\in L^q$, $p^{-1}+q^{-1}=1$ the dual pairing $\langle f,g\rangle$ is always finite, because of Hölder's inequality.