Kolmogoroff's criterion for standardization

The Normierbarkeitskriterium of Kolmogorov ( English Kolmogorov's standard ability criterion ) is a theorem of functional analysis , one of the branches of mathematics . It goes back to a work by the Russian mathematician Andrej Kolmogoroff from 1934.

criteria

Kolmogoroff's criterion for normalization states:

The topology of a Hausdorff topological vector space is generated by a norm if and only if its zero vector has a neighborhood which is a simultaneously bounded and convex subset of . ${\ displaystyle \ tau}$ ${\ displaystyle (E, \ tau)}$ ${\ displaystyle (E, \ tau)}$

If the mentioned condition is met, then there is a normalizable space . ${\ displaystyle (E, \ tau)}$

Application example

The above characterization of normalizable spaces can be used to determine that a room cannot be normalized:

The episode room

{\ displaystyle \ omega = \ prod _ {n \ in \ mathbb {N}} {\ mathbb {K}}}

all - sequences ( or ), provided with the product topology , is an infinite - dimensional, completely metrizable topological vector space, in which the zero sequence has no restricted neighborhood. Therefore it can not be normalized. ${\ displaystyle \ mathbb {K}}$ ${\ displaystyle \ mathbb {K} = \ mathbb {R}}$ ${\ displaystyle \ mathbb {K} = \ mathbb {C}}$ ${\ displaystyle (0) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle \ omega}$

Historical

Walter Rudin points out in his Functional Analysis (2nd edition, p. 400) that Kolmogoroff's criterion for normalization may be the first proposition of the theory of locally convex spaces .

literature

A. Kolmogoroff: On the normalizability of a general topological linear space . In: Studia Mathematica . tape 5 , 1934, pp. 29-33 ( matwbn.icm.edu.pl [PDF]).
Sterling K. Berberian: Lectures in Functional Analysis and Operator Theory (= Graduate Texts in Mathematics . Volume 15 ). Springer Verlag, Berlin (inter alia) 1974, ISBN 0-387-90080-2 , p. 55-56, 106-108 ( MR0417727 ).
Walter Rudin: Functional Analysis (= International Series in Pure and Applied Mathematics ). 2nd Edition. McGraw-Hill, Boston (et al.) 1991, ISBN 0-07-054236-8 , pp. 30, 400 ( MR1157815 ).
Dirk Werner : Functional Analysis (= Springer textbook ). Springer Verlag, Berlin (among others) 2007, ISBN 978-3-540-72533-6 , pp. 437 .