Intereting Posts

Does the sequence $(\sqrt{n} \cdot 1_{})_n$ converge weakly in $L^2$?
A problem about generalization of Bezout equation to entire functions
On Tarski-Knaster theorem
Proving Separation from Replacement
Do localization and completion commute?
Dilogarithm Inversionformula: $ \text{Li}_2(z) + \text{Li}_2(1/z) = -\zeta(2) – \log^2(-z)/2$
Ramanujan theta function and its continued fraction
Soft Question: Weblinks to pages with explanation on quadratics.
What is the limit of this sequence involving logs of binomials?
Equation of angle bisector, given the equations of two lines in 2D
Is a left invertible element of a ring necessarily right invertible?
Expected number of frog jumps
Global sections of a tensor product of vector bundles on a smooth manifold
Ring with four solutions to $x^2-1=0$
Axiomatic definition of sin and cos?

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why?

Any notes or suggestions will be appreciated.

- Proving without Zorn's Lemma: additive group of the reals is isomorphic to the additive group of the complex numbers
- $G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
- Computing easy direct limit of groups
- Subgroups of $\Bbb{R}^n$ that are closed and discrete
- Are these exactly the abelian groups?
- Structure Theorem for abelian torsion groups that are not finitely generated

- Find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)$ has order 2
- $F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$
- For abelian groups: does knowing $\text{Hom}(X,Z)$ for all $Z$ suffice to determine $X$?
- If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$
- The center of a non-Abelian group of order 8
- Need to prove that (S,*) defined by the binary operation a*b = a+b+ab is an abelian group on S = R \ {1}
- Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?
- Characterizing the cosets of a cycle of a finite abelian group with a linear combination of floor functions
- Does there exist any surjective group homomrophism from $(\mathbb R^* , .)$ onto $(\mathbb Q^* , .)$?
- Two term free resolution of an abelian group.

The stupid answer is that $\operatorname{Hom}_{\text{cts}}(\mathbb Z, S^1) = \operatorname{Hom}_{\mathbb Z}(\mathbb Z,S^1) = S^1$, so we are done by duality.

Here, $S^1$ denotes the unit circle $S^1 \cong \frac{\mathbb R}{2\pi\mathbb Z} \cong \mathbb R/\mathbb Z$, a.k.a. $\mathbb T$.

The real answer is that all characters of $S^1$ are of the form $z \mapsto z^n$. The reason is explained in many places, for example here: https://mathoverflow.net/questions/89504/quick-computation-of-the-pontryagin-dual-group-of-torus

In particular, you can see there that it is enough to know that $\mathbb R$ is selfdual.

- Does every Lebesgue measurable set have the Baire property?
- Textbooks on set theory
- What is the meaning of equilibrium solution?
- Dimension of $GL(n, \mathbb{R})$
- Summation of Fibonacci numbers $F_n$ with $n$ odd vs. even
- Disjoint closed sets in a second countable zero-dimensional space can be separated by a clopen set
- What are some applications outside of mathematics for algebraic geometry?
- Axiom of Choice: What exactly is a choice, and when and why is it needed?
- Right Inverse for Surjective Function
- Help in understanding the proof of Mersenne Prime
- Proving that every set $A \subset \Bbb N$ of size $n$ contains a subset $B \subset A$ with $n | \sum_{b \in B} b$
- The formula for a distance between two point on Riemannian manifold
- For two vectors $a$ and $b$, why does $\cos(θ)$ equal the dot product of $a$ and $b$ divided by the product of the vectors' magnitudes?
- subgroup generated by two subgroups
- What is combinatorics?