Topological Concepts — Weak Topologies


Coordinatewise convergence of sequences in the closed unit ball of l2 was already used by Hilbert in his definition of Vollstetigkeit; see The term schwache Konvergenz first appeared in Weyl’s thesis [1908, p. 8]:
Jedes bestimmte Wertsystem (x)=(x1,x2,...) werden wir einen Punkt unseres Raumes von unendlichvielen Dimensionen nennen und x1,x2,... bezw. seine 1.,2.,... Koordinate: \( x_1 = \mathfrak{C}\mathfrak{o}\left( x \right),x_2 = \mathfrak{C}\mathfrak{o}_2 \left( x \right) \),... . Haben wir eine unendliche Reihe solcher Punkte (x)1, (x)2,..., so sagen wir, sie konvergiere schwach gegen den Punkt (x), wenn für jeden Index i
$$ \mathop L\limits_{n = \infty } \mathfrak{C}\mathfrak{o}_i \left( x \right)^n = \mathfrak{C}\mathfrak{o}_i \left( x \right) $$
ist. We stress that Weyl did not assume boundedness, which means that in his sense (nen) would be a weak null sequence.


Banach Space Topological Space Linear Space Weak Topology Cluster Point 
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© Birkhäuser Boston 2007

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