Birkhoff-Kellogg's theorem
The set of Birkhoff Kellogg ( English Birkhoff Kellogg theorem ) is a theorem of the mathematical branch of nonlinear functional analysis , the a in 1922 by the two mathematicians George David Birkhoff and Oliver Dimon Kellogg presented scientific work back. It deals with the question of under what conditions for certain operators on infinite - dimensional Banach spaces , the eigenvalue problem is solved. The theorem turns out to be an analogue of the classical Poincaré-Brouwer theorem in topology .
Formulation of the sentence
The sentence can be summarized as follows:
- Given a infinite - dimensional Banach space and in a limited open subset which the zero contained .
-
On the closed envelope of a compact ( linear or nonlinear) operator is given that satisfies the condition
- .
- meet.
- Then:
- The eigenvalue problem is solvable for. There is an edge point and a real number that satisfy the equation .
background
The proof of the Birkhoff-Kellogg theorem is essentially based on a general eigenvalue principle , for the derivation of which the Leray-Schauder degree of mapping is used, as well as the following approximation theorem for compact operators ( English Approximation Theorem for Compact Operators ):
- Given are two Banach spaces (over with ) as well as a bounded, non-empty subset and then an arbitrary operator .
- Then:
-
is a compact operator if and only if there is a sequence of operators such that the following three conditions are always fulfilled:
- (i) is compact.
- (ii) .
- (iii) The of the image set (above ) spanned linear subspace has finite dimension .
literature
- GD Birkhoff, OD Kellogg: Invariant points in function space. In: Transactions of the American Mathematical Society . tape 23 , 1922, pp. 96-115 ( MR1501192 ).
- Eberhard Zeidler : Lectures on nonlinear functional analysis I: Fixed point theorems . BG Teubner Verlagsgesellschaft , Leipzig 1976 ( MR0473927 ).
- Eberhard Zeidler: Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems . Translated by Peter R. Wadsack. Springer Verlag , New York, Berlin, Heidelberg, Tokyo 1986, ISBN 0-387-90914-1 ( MR0816732 ).
Individual evidence
- ↑ Eberhard Zeidler: Lectures on nonlinear functional analysis I 1976, p. 12, pp. 152–153
- ^ Eberhard Zeidler: Nonlinear Functional Analysis and its Applications I 1986, p. 557 ff
- ^ Zeidler (1976), p. 153
- ^ Zeidler (1986), p. 559
- ↑ is the set of boundary points of .
- ↑ Zeidler (1976), p. 25, pp. 152-153
- ^ Zeidler (1986), p. 55, pp. 558-559