Differential Linear Logic extends Linear Logic by allowing the differentiation of proofs. Trying to interpret this proof-theoretical notion of differentiation by traditional analysis, one faces the fact that analysis badly accommodates with the very basic layers of Linear Logic. Indeed, tensor products are seldom associative and spaces stable by double duality enjoy very poor stability properties. In this work, we unveil the polarized settings lying beyond several models of Differential Linear Logic. By doing so, we identify chiralities - a categorical axiomatic developed from game semantics - as an adequate setting for expressing several results from the theory of topological vector spaces. In particular, complete spaces provide an interpretation for negative connectives, while barrelled or bornological spaces provide an interpretation for positive connectives.