Abstract
This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (non-translation invariant) valuations on a linear spaceV. We present three descriptions of this space. We describe the canonical multiplicative structure on this space generalizing the results from [4] obtained for polynomial valuations.
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