Orlicz–Pettis theorem

The set of Orlicz - Pettis (after Władysław Orlicz and Billy James Pettis ) is a set from the mathematical branch of functional analysis. It allows in a given situation of the weak convergence on the norm convergence to close in Banach spaces.

In infinite-dimensional Banach spaces the weak topology is strictly slower than the norm topology. For example, if the -th basis vector in Hilbert space, that is, that sequence a 1 and a 0 in all other places took part in the - th position, so the sequence converges with respect to the weak topology compared to 0, each continuous linear functional has namely the shape of a, and is therefore considered by the Riesz representation theorem. However, the result can not converge in norm, as a possible standard limit would also be 0, but it applies to all indices.

For series in Banach spaces, the situation looks the same. If in the above example and so is. Therefore, the series converges in the weak topology ( to 0 ), but not in the standard topology.

A series is teilreihenkonvergent when each part series converges, that is, when converged for each sequence. The difference described between weak convergence and norm convergence for teilreihenkonvergente rows no longer exists, that is precisely the content of the presented sentence:

Orlicz - Pettis set of:

This theorem was proved first in 1929 by Orlicz and independently in 1938 by Pettis. Modern proofs use the Bochner integral. Conversely, the vector-valued integration theory is precisely the motivation for Pettis to deal with this set. This set has experienced a number of generalizations, this is called Orlicz - Pettis sets the type. So, for example, applies in locally convex spaces that teilreihenkonvergenten rows coincide with respect to the weak topology and with respect to the Mackey topology.