So far, mathematical analysis has not been much in focus of lab; this could be compensated partly by noting that web (including wikipedia) has much more resources in the analysis area (similarly holds for combinatorics and number theory, cf. e.g. Tao’s blog), than in category theory and homotopy theory, topology, geometry, algebra, cohomology, descent and sheaf theory, which are more in our primary focus.
Some of the lab entries related to mathematical analysis include nonstandard analysis, Weierstrass preparation theorem, Fourier transform, Pontrjagin dual, functional analysis, diffiety, analytic geometry, differential geometry, Legendre polynomial, dilogarithm, Hilbert space, Banach space, Banach algebra, topological vector space, locally convex space, operator algebras, Gelfand spectrum, measure space, Lebesgue space, Sobolev space, bounded operator, compact operator, Fredholm operator…and a book entry Handbook of analysis and its foundations. Many of the basic notions used in analysis courses are described in the general topological context if they belong there, e.g. compact space, continuous map, compact-open topology and so on.
Various smoothness concepts in geometry, rarely studied in standard courses of analysis, but sometimes relevant, were studied to fair extent (and sometimes with inovations) in the lab, see e.g. differential form, generalized smooth space, stratifold, Frolicher space, synthetic differential geometry, generalized smooth algebras, infinitesimal object and some graded and super analogues (supermanifold, NQ-supermanifold, integration over supermanifolds); some other concepts of smoothness are rather algebraic and are used almost exclusively in the nonanalytic setups, e.g. formal smoothness of Grothendieck; see also algebraic approaches to differential calculus and tangent vector. Special attention has been paid to smooth group like objects like Lie group, Lie groupoid and their superanalogues and categorifications, as well as to their tangent structures like Lie algebroids and their interrelations (Lie theory, integration, Lie integration). Some other entries are related to the conceptual and categorical understanding of Feynman integral however so far from physical, conceptual and formal point of view only. This is closely related to understanding various spaces of sections in geometry and in study of sigma-models in physics (cf. space and quantity, geometric quantization, geometric function theory).
More material and suitable links are welcome.