The Topological Support of Gaussian Measure in Banach Space

The main result of the present paper is the theorem 1, which describes the topological support of an arbitrary Gaussian measure in a separable Banach space. This theorem will be proved after some discussion of the notion of support itself. But we begin with the reminder of the notion of covariance operator of a probability measure. This notion has a great importance not only for the description of support of Gaussian measures but also for the study of other problems in the theory of probability measures in linear spaces (c.f. [1], [2]). Let X be a real Banach space with topological dual X* and //bea probability measure on the σ-algebra generated by X*. We suppose that μ has the second order in weak sense, i.e. X* c L2(X,μ) (this restriction is necessary if we want to speak about an analogue of variance). Covariance operator R of measure μ is defined by the relation (see [1])