Review on Capacity of Gaussian Channel with or without Feedback

Let X be a real separable Banach space and X * be its dual space. Let B(X) be Borel σ-field of X. For finite dimensional subspace F of X * we define the cylinder set C based on F as follows C = {x ∈ X; It is easy to show that C(X, X *) is a fileld. Let C(X, X *) be the σ− field generated by C(X, X *). Then C(X, X *) = B(X). If µ is a probability measure on (X, B(X)) satisfying ∫ X ∥x∥ 2 dµ(x) < ∞, then there exist a vector m ∈ X and an operator ∫ X ⟨x, x * ⟩dµ(x), ⟨Rx * , y * ⟩ = ∫ X ⟨x − m, x * ⟩⟨x − m, y * ⟩dµ(x), for any x * ∈ X * , y * ∈ Y *. m is a mean vector of µ and R is a covariance operator of µ which is a bounded linear operator. We remark that R is symmetric in the following sence.