In this thesis, we consider the problem of capturing finite-representability between Banach spaces using the tools of continuous model theory. We introduce predicates and additional sorts to capture finite-representability and show that these can be used to expand the language of Banach spaces. We then show that the class of infinite-dimensional Banach spaces expanded with this additional structure forms an elementary class K_G , and conclude that the theory T_G of K_G is interpretable in T^{eq} , where T is the theory of infinite-dimensional Banach spaces. Finally, we show that existential equivalence in a reduct of the language implies finite-representability. Relevant background on continuous model theory and Banach space theory is provided. / Thesis / Master of Science (MSc)

<p>In this thesis we prove a strong conceptual completeness result for first-order continuous logic. Strong conceptual completeness was proved in 1987 by Michael Makkai for classical first-order logic, and states that it is possible to recover a first-order theory T by looking at functors originating from the category Mod(T) of its models. </p> <p> We then give a brief account of simple theories in continuous logic, and give a proof that the characterization of simple theories using dividing holds in continuous structures. These results are a specialization of well established results for thick cats which appear in [Ben03b] and in [Ben03a].</p> <p> Finally, we turn to the study of non-archimedean Banach spaces over non-trivially valued fields. We give a natural language and axioms to describe them, and show that they admit quantifier elimination, and are N0-stable. We also show that the theory of non-archimedean Banach spaces has only one N 1-saturated model in any cardinality. </p> / Thesis / Doctor of Philosophy (PhD)