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The compact-open character group is a Hausdorff topological abelian group

Statement

Let $G$ be an abelian topological group and let $\hat{G}$ be its Pontryagin dual (The Pontryagin dual with the compact-open topology). With pointwise multiplication and inversion, $\hat{G}$ is a Hausdorff topological abelian group (Topological group: multiplication and inversion are continuous, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not). Explicitly, for $f, g \in \hat{G}$ , compact $K \subseteq G$ and $ε > 0$ , writing $U_{K} (f, r) : = {h \in \hat{G} : ∣ h (x) - f (x) ∣ < r for every x \in K}$ for $r > 0$ , these sets form a neighbourhood basis at $f$ , and $U_{K} (f, ε / 2) \cdot U_{K} (g, ε / 2) \subseteq U_{K} (f g, ε), U_{K} (f, ε)^{- 1} = U_{K} (f^{- 1}, ε) .$

Facts & Assumptions

[F1]

$T$ is a compact metrizable topological abelian group; in particular multiplication and inversion are continuous and every element has modulus $1$ . For all $z, w \in T$ one has $∣ z^{- 1} - w^{- 1} ∣ = ∣ z - w ∣$ and $∣ z w - z ∣ = ∣ w - 1 ∣$ . (The multiplicative unit circle is a compact metrizable topological abelian group)

[F2]

$\hat{G} = {Hom}_{c t s} (G, T)$ is the set of continuous homomorphisms $γ : G \to T$ , with pointwise multiplication and the compact-open topology with subbasis $S (K, V) = {γ : γ [K] \subseteq V}$ for compact $K \subseteq G$ and open $V \subseteq T$ . (The Pontryagin dual with the compact-open topology)

[F3]

For all complex $z, w$ : $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ , $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ , and $z \overline{z} = ∣ z ∣^{2}$ ; also $∣ \exp (i y) ∣ = 1$ for real $y$ . (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ )

[F4]

A map into a product space is continuous exactly when all its components are, and composites of continuous maps are continuous. (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous)

[F5]

Subbasic open sets on a subspace are the traces of subbasic open sets of the ambient space, and singletons are compact. (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, The compact-open topology on $C (X, Y)$ for arbitrary topological spaces, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right)

[F6]

$T$ is Hausdorff: distinct points of a metric space are separated by disjoint open balls. (Distinct points of a metric space have disjoint balls around them, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not)

[F7]

Continuous images of compact sets are compact; continuous real-valued functions on nonempty compact spaces attain their maxima; closed subsets of compact spaces and finite unions of compact subsets are compact. Compact subsets admit finite subcovers from ambient open covers. (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact, A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it)

Proof

Given: An abelian topological group $G$ and its dual $\hat{G}$ with the compact-open topology.

1.1F1F2F4

Pointwise multiplication and inversion are well defined on $\hat{G}$ and give it the structure of an abelian group: for $f, g \in \hat{G}$ the maps $f g$ and $f^{- 1}$ are group homomorphisms because $(f g) (x + y) = f (x + y) g (x + y) = f (x) f (y) g (x) g (y) = f (x) g (x) f (y) g (y) = (f g) (x) (f g) (y)$ and $f^{- 1} (x + y) = f (x + y)^{- 1} = f (x)^{- 1} f (y)^{- 1}$ by [F2] and commutativity of $T$ ; they are continuous because $x \mapsto (f (x), g (x))$ is continuous into the product by [F4], multiplication on $T$ is continuous by [F1], and $f g$ is the composite of these two maps, while $f^{- 1}$ is the composite of $f$ with the continuous inversion of $T$ by [F1] and [F4]. The group axioms for $\hat{G}$ hold pointwise because $T$ is an abelian group by [F1], with pointwise constant $1$ as identity.

1.2F2F3F7

Each $U_{K} (f, r)$ is compact-open open. First, for any character $h$ and $a > 0$ , cover the compact image $h [K]$ by finitely many balls $B (z_{j}, a / 3)$ with centres in $T$ , and put $L_{j} : = {x \in K : ∣ h (x) - z_{j} ∣ \leq a / 3}$ . These closed subsets of $K$ are compact and cover $K$ by [F7]. The open set $W_{h} : = ⋂_{j} S (L_{j}, B (z_{j}, 2 a / 3))$ contains $h$ and lies in $U_{K} (h, a)$ by the triangle inequality. Now if $h \in U_{K} (f, r)$ and $K \neq \emptyset$ , the continuous function $x \mapsto ∣ h (x) - f (x) ∣$ attains a maximum $m < r$ by [F7]; its continuity follows from $∣ ∣ u - v ∣ - ∣ u^{'} - v^{'} ∣ ∣ \leq ∣ u - u^{'} ∣ + ∣ v - v^{'} ∣$ . Choose $0 < a < r - m$ . The preceding $W_{h}$ is contained in $U_{K} (f, r)$ , so every member of $U_{K} (f, r)$ has an open neighbourhood inside it. If $K = \emptyset$ , $U_{K} (f, r) = \hat{G}$ .

1.3F1F3

The displayed estimates hold: for $f^{'} \in U_{K} (f, ε / 2)$ , $g^{'} \in U_{K} (g, ε / 2)$ and $x \in K$ , $∣ f^{'} (x) g^{'} (x) - f (x) g (x) ∣ \leq ∣ f^{'} (x) - f (x) ∣ + ∣ g^{'} (x) - g (x) ∣ < ε$ , since all values have modulus $1$ . Also $∣ f^{'} (x)^{- 1} - f (x)^{- 1} ∣ = ∣ f^{'} (x) - f (x) ∣$ ; inversion is involutive, so the second displayed equality follows.

1.4F2F5F6

$\hat{G}$ is Hausdorff: if $f \neq g$ in $\hat{G}$ , there is $x \in G$ with $f (x) \neq g (x)$ ; by [F6] choose disjoint open $V, W \subseteq T$ with $f (x) \in V$ , $g (x) \in W$ ; then $S ({x}, V)$ and $S ({x}, W)$ are open in $\hat{G}$ by [F5], they contain $f$ and $g$ respectively, and they are disjoint because no function can take the same value in both $V$ and $W$ .

2.1step 1.2F2F3F7

These sets form a neighbourhood basis at $f$ . If $f \in S (K, V)$ , cover $f [K]$ by finitely many balls $B (z_{j}, a_{j})$ with $a_{j} > 0$ and $B (z_{j}, 2 a_{j}) \subseteq V$ , using compactness of $f [K]$ and openness of $V$ . For $r : = \min_{j} a_{j} > 0$ , the triangle inequality gives $U_{K} (f, r) \subseteq S (K, V)$ . For empty $K$ take any $r > 0$ . Any finite intersection of such subbasic neighbourhoods contains $U_{⋃_{j} K_{j}} (f, \min_{j} r_{j})$ , and the union is compact by [F7]; an empty intersection is the whole dual. Together with step 1.2 this proves the basis assertion.

3.1step 1.1step 2.1step 1.3F4

Multiplication and inversion on $\hat{G}$ are continuous. By step 2.1 it suffices to test $U_{K} (f g, ε)$ and $U_{K} (f^{- 1}, ε)$ at arbitrary $f, g$ . Step 1.3 maps the open rectangle $U_{K} (f, ε / 2) \times U_{K} (g, ε / 2)$ into the first set and maps the open neighbourhood $U_{K} (f, ε)$ into the second.

4.1step 1.1step 2.1step 1.3step 1.4step 3.1∎

The pointwise group of step 1.1 is Hausdorff by step 1.4 and has continuous operations by step 3.1, so it is a Hausdorff topological abelian group. The asserted neighbourhood basis and estimates are steps 2.1 and 1.3.

Depends on

Used by

Cited to discharge well-definedness by The Pontryagin dual with the compact-open topology.

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Sources

Dikran D. Dikranjan, Introduction to Topological Groups (author lecture notes, Universita di Udine / Universidad Complutense de Madrid, 2007) (standard reference, not scraped)
Lynn H. Loomis, Introduction to Abstract Harmonic Analysis, D. Van Nostrand 1953, Chapter VII, Sections 34-35 (printed pp. 134-140) (standard reference, not scraped)