Linearization of proper actions of locally compact groups on Tychonoff spaces

We prove that if G is a locally compact group acting properly (in the sense of R. Palais) on a Tychonoff space X, then X can be embedded equivariantly into a linear G-space L endowed with a linear G-action which is proper on the complement L∖{0}. If, in addition, G is a Lie group and τ an infinite cardinal number, then the linearizing G-space L can be chosen to be the same for all proper G-spaces X of weight w(X)⩽τ.