Following an introductory chapter on constructive mathematics, Chapter 2 contains a detailed constructive analysis of the Toeplitz-Hausdorff Theorem on the convexity of the numerical range of an operator in a Hilbert space. It is shown that the results in the chapter are the best possible with constructive methods. The rest of the thesis deals with the constructive theory of not-necessarily-commutative Banach algebras. Chapter 3 discusses the Spectral Mapping Theorem in that context, again showing that the results obtained are the best possible. Chapter 4 deals with the question, "Are positive integral powers of a hermitian element of a Banach algebra hermitian?" A major problem that has to be overcome is to find the 'right' constructive definition of hermitian, since there is no guarantee in constructive mathematics that the state space of a Banach algebra is nonempty; this forces us to work with approximations to the state space, rather than the state space itself. In the final chapter, these approximations are used to give careful estimates that lead us to a proof of Sinclair's Theorem that the spectral radius of a hermitian element equals its norm. The thesis has two appendices: one describing the axioms of intuitionistic first order logic, and the other giving a proof of the Spectral Theorem for normal operators on a separable Hilbert space.