I am reading LECTURE NOTES 2 FOR 247A,TERENCE TAO. At the beginning of the paragraph the author defines the fourier trasform $\mathcal{F}f$ of a function $f$ and the hypothesis is that $f \in L^1(\mathbb{R}^{d})$. Then at pg. $19$ he says that for a function $g \in L^1(\mathbb{R}^{2d})$ one has $$\langle f, g \rangle =\langle \mathcal{F}f, \mathcal{F}g \rangle.$$
The author defines $\langle \cdot, \cdot \rangle$ in LECTURE NOTES 1 FOR 247A, TERENCE TAO pg. $28$ formula $(20)$.
Is it right to interpret $\langle \cdot, \cdot \rangle $ as the classical pairing between a space $X$ and its dual $X'$?
If yes, does this means that If I suppose additionally $f \in L^p(\mathbb{R}^{d})$ and $g \in L^{p'}(\mathbb{R}^d), \quad \dfrac{1}{p}+\dfrac{1}{p'}=1$, the following Parseval identity holds $$\int_{\mathbb{R}^d} f(x)g(x)dx = \int_{\mathbb{R}^d} \mathcal{F}f(\xi) \mathcal{F}g (\xi) d\xi \quad ? $$