A method is here given for modelling the Anderson-Belnap logics of positive entailment (LE) and relevant implication (LR) in established mathematical structures, notably topological spaces and finite- dimensional Banach spaces, respectively. For topological spaces four constructions are required: Function spaces, \(Y^ X\), with pointwise convergence, Cartesian product spaces, \(X\times Y\), with product topology, Cartesian product spaces, \(X\otimes Y\), with tensor product topology, and disjoint union spaces, \(X+Y\), with coproduct topology, as well as fixed spaces with one point, 1, which model entailment, extensional conjunction, intensional conjunction, disjunction, and the propositional constant, I, of (LE). From these functions realizing the axioms of (LE) are defined according to their derivations in a Gentzen- style consecution calculus for (LE). Since possession of realizing functionals is preserved by the rules of (LE), a formula, A, is provable in (LE) iff there exists a realizing functional \(f(A):1\to {\mathfrak M}(A),\) for all appropriate models \({\mathfrak M}\) of (LE). For the full logic of relevance (LR) the constructions are similar, where 1 is replaced by \({\mathbb{R}}\), the space of real numbers, with the result that a formula is a theorem of (LR) iff it is realizable by an explicitly definable continuous linear transformation in the category of finite-dimensional Banach spaces and linear contractions.