In functional analysis, the Birkhoff–Kellogg invariant-direction theorem, named after G. D. Birkhoff and O. D. Kellogg, is a generalization of the Brouwer fixed-point theorem. 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.
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