- (1)
$\wh \real \iso \real$ with dual pairing $\dpn{x, \xi}{\real}= e^{2\pi i \xi x}$.
- (2)
$\wh \mathbb{T}\iso \integer$ with dual pairing $\dpn{x, n}{\mathbb{T}}= x^{n}$.
- (3)
$\wh \integer = \wh \mathbb{T}$ with dual pairing $\dpn{n, x}{\mathbb{T}}= x^{n}$.
Proof, [Proposition 4.6, Fol16]. (1): Let $\phi \in \wh \real$, then $\dpn{0, \phi}{\real}= 1$. Thus there exists $\alpha > 0$ with $A := \int_{0}^{\alpha} \dpn{x, \phi}{\real}dx \ne 0$. In which case,
\[A \dpn{x, \phi}{G}= \int_{0}^{\alpha} \dpn{x + y, \phi}{\real}dy = \int_{x}^{x + \alpha}\dpn{y, \phi}{\real}dy\]
so $\phi \in C^{1}(\real)$ and there exists $c \in \complex$ with $\phi' = c\phi$. Therefore $\dpn{x, \phi}{\real}= e^{cx}$ for all $x \in \real$.$\square$
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