Complex symplectic spaces, defined earlier by the authors in their AMS Monograph, are non-trivial generalizations of the real symplectic spaces of classical analytical dynamics. These spaces can also be viewed as non-degenerate indefinite inner product spaces, although the authors here follow the lesser known exposition within complex symplectic algebra and geometry, as is appropriate for their prior development of boundary value theory. In the case of finite dimensional complex symplectic spaces it was shown that the corresponding symplectic algebra is important for the description and classification of all self-adjoint boundary value problems for (linear) ordinary differential equations on a real interval. In later AMS Memoirs infinite dimensional complex symplectic spaces were introduced for the analysis of multi-interval systems and elliptic partial differential operators.

In this current Memoir the authors present a self-contained, systematic investigation of general complex symplectic spaces, and their Lagrangian subspaces, regardless of the finite or infinite dimensionality—starting with axiomatic definitions and leading towards general Glazman-Krein-Naimark (GKN) theorems. In particular, the appropriate relevant topologies on such a symplectic space \(\mathsf{S}\) are compared and contrasted, demonstrating that \(\mathsf{S}\) is a locally convex linear topological space in terms of the symplectic weak topology. Also the symplectic invariants are defined (as cardinal numbers) characterizing \(\mathsf{S}\), in terms of suitable Hilbert structures on \(\mathsf{S}\).

The penultimate section is devoted to a review of the applications of symplectic algebra to the motivating of boundary value problems for ordinary and partial differential operators.

The final section, the Aftermath, is a review and summary of the relevant literature on the theory and application of complex symplectic spaces.

The Memoir is completed by symbol and subject indexes.