Accattoli, Dal Lago, and Vanoni have recently proved that the space used by the \textit{Space KAM}, a variant of the Krivine abstract machine, is a reasonable space cost model for the $\lambda$-calculus accounting for logarithmic space, solving a longstanding open problem. In this paper, we provide a new system of \textit{multi types} (a variant of intersection types) and extract from multi type derivations the space used by the Space KAM, capturing into a type system the space complexity of the abstract machine. Additionally, we show how to capture also the time of the Space KAM, which is a reasonable time cost model, via minor changes to the type system.