Vol. 38, No. 1, 1971

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Two uniform boundedness theorems

James DeWitt Stein

Vol. 38 (1971), No. 1, 251–260

A geodesically convex space is a metric space in which each two points can be connected by a unique segment (a path of minimal length). An affine transformation between two geodesically convex spaces is a map which takes segments into segments. It is shown that, if the domain is complete, a pointwise-bounded family of continuous affine transformations is uniformly bounded. Under a mild additional hypothesis, the following stronger theorem holds: if

ℱ = {Tσ|A ∈ A }

is a pointwise-bounded family of affine transformatons and Ta is continuous on a closed geodesically convex Sα with

Sα ⁄= ∅,

then α1,n such that 𝒯 is uniformly bounded on

S αk.

Mathematical Subject Classification 2000
Primary: 52A50
Secondary: 54H20
Received: 17 March 1970
Revised: 14 April 1971
Published: 1 July 1971
James DeWitt Stein