Orlicz–Pettis theorem
Encyclopedia
In functional analysis
, the Orlicz–Pettis theorem is a theorem about convergence in Banach space
s. It is named for Władysław Orlicz and Billy James Pettis
. The result was originally proven by Orlicz for weakly sequentially complete normed spaces.
be any sequence in X. If the series
is weakly subseries convergent, then the series is actually subseries convergent in the norm topology of X.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the Orlicz–Pettis theorem is a theorem about convergence in Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s. It is named for Władysław Orlicz and Billy James Pettis
Billy James Pettis
Billy James Pettis , was an American mathematician, known for his contributions to functional analysis.-See also:*Dunford–Pettis property*Dunford–Pettis theorem*Milman–Pettis theorem*Orlicz–Pettis theorem*Pettis integral...
. The result was originally proven by Orlicz for weakly sequentially complete normed spaces.
Orlicz–Pettis theorem for normed spaces
Let X be a Banach space and let be any sequence in X. If the seriesSeries (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
is weakly subseries convergent, then the series is actually subseries convergent in the norm topology of X.