In
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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, the Birkhoff–Kellogg invariant-direction theorem, named after
G. D. Birkhoff and
O. D. Kellogg, is a generalization of the
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest ...
. The theorem states that:
Let ''U'' be a bounded open neighborhood of 0 in an infinite-dimensional normed linear space ''V'', and let ''F'':∂''U'' → ''V'' be a compact map satisfying , , ''F''(''x''), , ≥ α for some α > 0 for all ''x'' in ∂''U''. Then ''F'' has an invariant direction, ''i.e.'', there exist some ''x''
o and some ''λ'' > 0 satisfying ''x''
o = ''λF''(''x''
o).
The Birkhoff–Kellogg theorem and its generalizations by
Schauder and
Leray have applications to partial differential equations.
References
Theorems in functional analysis
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