Banach–Alaoglu theorem

The Banach - Alaoglu (also set by Alaoglu or set of Alaoglu - Bourbaki or in a more general version of Banach - Alaoglu - Bourbaki ) is a compact set and is generally associated with the field of functional analysis, although it has a purely topological statement and contains substantially follows from the set of Tychonoff.

The set

It should be a normed space. Then the amount of

Compact in the weak -* topology, where the topological dual space of designated.

Discussion

The meaning of this statement derives mainly from the comparison with the lemma of Riesz, after which the norm closed unit ball of a normed space if and is compact in the norm topology, if the space has finite dimension. The topological dual space, ie the space of all continuous linear functionals on a normed space is itself normalized again by virtue

The norm- closed unit ball in just the quantity. With is also of infinite vector space dimension. Used follows from Lemma Riesz, that is in the case of non-standard compact. But well is compact in the weak weak -* topology.

It should be noted again at this point that is used to construct the standard of the compactness but does not apply in the norm topology, but in the weak -* topology.

In connection with the above comparison can also be the classification of the set of Banach Alaoglu in the area of functional analysis justified, because only at infinite dimension of the underlying normed space, the statement is not trivial ( and with the above standard are in the finite topologically isomorphic, and the weak -* topology is equal to the norm topology ).

Note that the set of Banach Alaoglu does not imply local compactness of the weak -* topology, because it is coarser than the norm topology and the closed unit ball no zero environment. Every locally compact topological vector space is in fact finite.

Application

Compact sets are in the ( functional ) analysis is always of great importance. Since they in infinite-dimensional normed spaces (according to the above lemma Riesz and generalized non - local compactness ) are rather scarce, the change to the weaker * - topology but in many situations is not a big restriction and this topology in a natural way into the game comes, gives this set a plethora of "new" compact sets to the hand. A prominent example is to be mentioned here the proof of the theorem of Gelfand - Neumark from the theory of C *-algebras, which establishes an isometric isomorphism between any commutative C * - algebra and the continuous functions on a compact set. The compactness of the crowd follows it from an application of the theorem of Banach - Alaoglu.

Generalizations and other formulations

Generalization: set of Alaoglu - Bourbaki

The Banach - Alaoglu can be formulated for more general topological vector spaces.

Be a locally convex space. For a neighborhood of zero in is

( the so-called polar of ) a - compact set.

For Banach spaces

The unit ball in the dual space of a Banach space is weak -* compact.

For separable Banach spaces

The unit ball in the dual space of a separable Banach space is compact with the weak -* topology and weak - * - metrizable, which is why they so is also weak -* sequentially compact. That a sequence has a weak - * convergent subsequence with limit in.