Hilbert space splittings: Abstract theory

This chapter focuses on the study of the the relation between (𝑉; 𝑎), namely a separable Hilbert space 𝑉 with scalar product 𝑎(•, •), and what we call a space splitting (V; a;R; 𝜌) = (𝑉𝑖; 𝑎𝑖; 𝑅𝑖; 𝜌𝑖)𝑖∈Ω, Ω ⊂ ℕ, associated with (𝑉; 𝑎) at an abstract level. The notion of stability, the role of the Schwarz operator associated with a space splitting, and operations on space splittings are discussed.