HilbertRaumbaum

HilbertRaumbaum is a theoretical construct that combines ideas from Hilbert space theory with a hierarchical, tree-based decomposition. In this framework, a rooted, countably branching tree T is assigned a family of separable Hilbert spaces {H_v} to its nodes v, together with bounded linear embeddings from child spaces into their parent spaces. The root carries a distinguished global space that serves as the ambient context for the entire construction. The global HilbertRaumbaum H is formed as the direct limit of the system defined by these embeddings, yielding a complete inner product space that reflects both the local subspaces and their hierarchical connections. The tree structure encodes a multi-scale decomposition in which subspaces at deeper levels represent finer scales and their coherence with higher levels is maintained via the embeddings and associated projections.

Properties of HilbertRaumbaum include separability and completeness under standard assumptions (for example, if each H_v is separable and T is countable). Projections onto the subspaces H_v interact with the tree in a coherent way: the projection associated with a parent node is determined by the contributions of its children through the embeddings. The construction supports natural filtrations by tree level, yielding a sequence of subspaces that converges to the global space in the limit.

Examples include simple finite-depth trees with finite-dimensional node spaces, which produce finite-dimensional approximations of the global structure, and infinite, orthogonally embedded trees that resemble wavelet-type multi-scale decompositions in an abstract setting.

History and usage: HilbertRaumbaum appears primarily in speculative mathematical discussions and in fiction that explores hierarchical representations in infinite-dimensional spaces. It is not a standard concept in established mathematical literature, and references to it are typically fictional or pedagogical. See also Hilbert space, direct limit, multi-resolution analysis, hierarchical modeling.