I am reading "An introduction to harmonic analysis" written by Y. Katznelson and I have a problem with understanding his symbol for scalar product of a functional and a function. This is my problem:
He define a homogeneous Banach space on $\mathbb{T}$ as a linear subspace $B$ of $L^1(\mathbb{ T})$ with a norm $\|\quad \|_B \geq \| \quad\| _ {L^1}$ under which it is a Banach space and having the following properties:
(H-1) If $f\in B$ and $\tau \in \mathbb{T},$ then $f_\tau \in \mathbb {B}$ and $\|f_\tau\|_B = \|f\|_B,$
(H-2) For all $f \in B \quad \lim_{\tau \to \tau_0} \|f_\tau - f_{\tau_0}\|_B = 0.$
After that, he introduces the Fourier coefficients of a functional $\mu \in B^*$ as a number $\overline{\langle e^{int} , \mu \rangle}$(with assume that $e^{int} \in B$ for every $n \in \mathbb{Z}).$ This is unfamiliar to me, because in all books I have read from analysis I only seen scalar product with a functional on the first place. Somehow I figured out that must be $\langle f , \mu \rangle = \overline{\mu(\overline{f})},$ but that makes me a problem with proving that an adjoint operator( which is defined in this book as an operator which satisfy $\langle A f, \mu \rangle = \langle f, A^* \mu\rangle$) have the same norm as an operator $A: B \to B.$
Can someone show me where I am making a mistake?
The author takes $\langle f,\mu\rangle$ as the value of $\mu(f)$ where $f\in B$ and $\mu\in B^\ast$. If we suppose $e^{int}\in B$, then we can define $$ \hat\mu(n):=\overline{\langle e^{int},\mu\rangle}=\overline{\mu(e^{int})}\in\mathbb C. $$ It is just a convention to express the value of a linear functional on an element. Probably the reason is to emphasize the duality of $B$ and $B^\ast$. Since you can either view $ \mu(f) $ as the linear functional $\mu\in B^\ast$ evaluated at $f\in B$, or the linear functional $f\in B\hookrightarrow B^{\ast\ast}$ evaluated at $\mu\in B^\ast$ as long as we note that $$ |\mu(f)|\le\|\mu\|_{B^\ast}\|f\|_B $$ which implies that both $\mu$ and $f$ are bounded functionals.
Do not confuse this notation as the inner product in Hilbert spaces, though we also use $\langle x,y\rangle$ to denote the inner product of $x,y\in\mathcal H$ where $\mathcal H$ is a Hilbert space.
So it makes no sense to verify the norms are equal. $e^{int}$ and $\mu$ are just not in some Hilbert space at the same time.