This is the final version of the paper, which appeared previously as 0354.
We show how algebraic K-theory and related theories can be defined by "algebraic mapping spaces". We accomplish this by constructing certain Gamma-objects in the category of ind-schemes which provide geometric representations of these theories. For example, the algebraic K-theory of a smooth variety X is shown to arise as the simplicial set of maps (defined using the standard cosimplicial variety) from X to a Gamma-object formed out of certain Grassmann varieties. Also, the K'-theory of a projective variety Y is shown to arise as the simplicial set of maps from Spec k to a certain ind-scheme defined in terms of Y. Most generally, the K-theory of coherent sheaves on the product of X and Y which are flat over X and satisfy a dimension of support condition may be realized in this fashion.
As explained in the introduction, we foresee that our results ought to be useful to consider different topological realizations of K-theory and these related theories. This can be accomplished when the ground field is the complex numbers, for example, by taking the analytic realization of the Gamma-object representing the theory in question.