Von Neumann's Inequality : a proof based on Linear Algebra

Von Neumann's inequality asserts that, given a contraction T operator on a Hilbert space H, an upper bound may be found for the norm of p(T), where p is a polynomial with complex coefficients. This upper bound is equal to the maximum of z in the closed unit disk of the norm of p(z). The formulation of this theorem only requires elementary notions, yet the proof is usually approached through Functional Analysis. In this thesis this result is tackled utilising solely Linear Algebra, including a proof of the Maximum Modulus Principle. In addition, variations on this inequality will also be touched upon.