A linear functional φ‎ on a normed vector space V is bounded if there exists M such that |φ‎(v)| ≤ M∥v∥ for all vε‎V. The analytic dual space V’ is the space of all bounded linear functionals on V with pointwise addition and scalar multiplication. V’ is a Banach space, whether or not V is complete; the norm on V’ is ∥φ‎∥ = sup{|φ‎(v)| : ∥v∥ = 1}.