A proof of the Pontryagin
duality theorem for locally compact abelian (LCA) groups is given, using
category-theoretical ideas and homological methods. The proof is guided by the
structure within the category of LCA groups and does not use any deep results
except for the Peter-Weyl theorem. The duality is first established for the
subcategory of elementary LCA groups (those isomorphic with Ti⊕ Zj⊕ Rk⊕ F,
where T is the circle group, Z the integers, R the real numbers, and F a finite
abelian group), and through the study of exact sequences, direct limits and projective
limits the duality is expanded to larger subcategories until the full duality theorem is
reached.
