Mazur's Theorem (weak and strong convergence)

The set of Mazur (after Stanislaw Mazur ) is a set of the functional analysis , the a correlation between the weak and the strong convergence indicates. From the definitions it immediately follows that every strongly converging sequence also converges weakly, whereas the weak convergence is not a sufficient criterion for the strong convergence. Mazur's theorem states that one can construct a strongly convergent sequence from convex combinations of terms of a weakly convergent sequence.

Formulation of the sentence

${\displaystyle X}$be a normalized vector space and a sequence that converges to weakly . Then there exists a sequence of convex combinations of (i.e. with ), so that strongly (i.e. with respect to the norm of ) converges to. ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x\in X}$ ${\displaystyle y_{n}}$ ${\displaystyle x_{n}}$ ${\displaystyle \textstyle y_{n}=\sum _{k=1}^{N_{n}}\lambda _{k,n}x_{k}}$ ${\displaystyle \textstyle \sum _{k=1}^{N_{n}}\lambda _{k,n}=1,\lambda _{k,n}\geq 0}$ ${\displaystyle y_{n}}$ ${\displaystyle X}$ ${\displaystyle x}$

Evidence sketch

Two results from the functional analysis are required: (1) In locally convex topological vector spaces , closed and convex sets are weakly closed. (2) In addition, the norm closure of convex sets is convex again.

Every normalized vector space is a locally convex topological vector space.

So consider the set of all convex combinations of (the so-called convex hull). Its standard closure is again convex (2), so the closed convex envelope of the weakly closed (1). Now, as a weak limit value of elements from the closed convex hull, one element of this hull. This means that the limit of a sequence of convex combinations must be . ${\displaystyle x_{n}}$ ${\displaystyle x_{n}}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x_{n}}$

source

• Dirk Werner : Functional Analysis . 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , page 108