Bernstein-Markov theorem for normed spaces

Let X and Y be real normed linear spaces and let φ: X → R be a non-negative function satisfying φ(x + y) ≤ φ(x) + ∥y∥ for all x, y ∈ X. We obtain estimates for optimal constants, whose existence is proven, and present applications to polynomials and multilinear mappings in normed spaces.