What is missing? The algebraic structure of a normed space allowed us to operate with vectors (addition and scalar multiplication), and its topological structure (the one endowed by the norm) gave us a notion of closeness (by means of the metric generated by the norm), which interacts harmoniously with the algebraic operations. In particular, it provided the notion of length of a vector. So what is missing if algebra and topology have already been properly laid on the same underlying set? A full geometric structure is still missing. Algebra and topology are not enough to extend to abstract spaces the geometric concept of relative direction (or angle) between vectors that is familiar in Euclidean geometry. The keyword here is orthogonality, a concept that emerges when we equip a linear space with an inner product. This supplies a tremendously rich structure that leads to remarkable simplifications.
KeywordsHilbert Space Orthonormal Basis Product Space Unitary Transformation Partial Isometry
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