The Tannaka-Krein theorem (in the narrow sense) is a particular Tannaka duality theorem for compact topological groups.
The original theorem is (in two independent and different formulations) stated and proved in
Tadao Tannaka, Über den Dualitätssatz der nichtkommutativen topologischen Gruppen, Tohoku Math. J. 45 (1938), n. 1, 1–12 (project euclid has only Tohoku new series!)
M.G. Krein, A principle of duality for bicompact groups and quadratic block algebras, Doklady AN SSSR 69 (1949), 725–728.
For a textbook treatment see
E. Hewitt, K. A. Ross, Abstract harmonic analysis, 2 vols.