Exercise 12, § 2

On uses the notation L(G) of exercise 11.

(a) Let H be a closed subgroup of G. Every element of L(H) extends to an element of L(G). (First reduce to the case where $G =\mathbf{R}^n\times D$ with D discrete, then to the case where the given element of L(H) is trivial on $\mathbf{R}^n$, then to the case where G is discrete. Use then the fact that $\mathbf{R}$ is divisible.)

(b) Every continuous morphism from H into $\mathbf{R}^n\times \mathbf{T}^I$, where I is any set, extends to a continuous morphism from G into $\mathbf{R}^n\times \mathbf{T}^I$. (Use a) and th. 4 of II, p. 226.)

(c) Conversely, let A be a locally compact commutative group. Suppose that, for every locally compact commutative group G, for every closed subgroup H of G, and every continuous morphism $\varphi$ from H into A$,\varphi$ extends to a continuous morphism from G into A. Then there exists an integer $n$ and a set I such that A is isomorphic to $\mathbf{R}^n\times \mathbf{T}^I$. (Taking $G =\mathbf{R},H =\mathbf{Z}$, one sees that A is connected, hence of the form $\mathbf{R}^n\times \widehat{D}$ with D discrete. Prove that D is a $\mathbf{Z}$-projective module, hence free.)