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### Conceptual explanation for Poisson summation formula

The Poisson summation formula says that for a Schwartz function $f : \mathbf R^d \to \mathbf R$ and its Fourier transform $\widehat f$, we have
$$\sum_{n \in \mathbf Z^d} f(x) = \sum_{n \in \mathbf ...

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### Creating Duals in A Category

Before stating my question I would like to provide afew motivating examples:
Examples:
In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom_R(M,R)$ satisfies: ...

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### Bohr compactification and “discretization”

Let $G$ is a compact group. We can form the Pontriagin dual $\widehat{G}$ of $G$: it is then discrete space. One can consider the Bohr compactification $b\widehat{G}$ of $\widehat{G}$ which is compact ...

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### What is the dual of a pre-injective map?

In [M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math.
Soc. (JEMS) 1 (1999), 109–197], Gromov introduces the notion of pre-injective map. Recasting this notion in the setting of ...

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### Proof that the Pontryagin dual of a topological group is a topological group

I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* ...

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### Quick computation of the Pontryagin dual group of torus

I'm looking for a quick way to compute the Pontryagin dual group of the n-dimensional torus $\mathbb{T}^n$ (with $\mathbb{T} := \mathbb{R} / \mathbb{Z}$). The only way I know is from "Dikran Dikranjan ...

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### Discrete-compact duality for nonabelian groups

A standard property of Pontrjagin duality is that a locally compact Hausdorff abelian group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for ...

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### Fourier Transform of measure on Banach Space (a question about Pontryagin Duality)

The following definition is given as the Fourier transform of a Borel probability measure $\mu$ on $E$, a Banach Space (Real):
$\hat{\mu}: E^*\rightarrow \mathbb{C}$ defined by
...

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### Fourier transforms of compactly supported functions

One manifestation of the uncertainty principle is the fact that a compactly supported function $f$ cannot have a Fourier transform which vanishes on an open set. As stated, this phenomenon applies ...

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### Injective modules and Pontrjagin duals

Forgive me for this naive question.
We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, page 784.
Every module is a submodule of an ...