Banach space

A Hilbert space gives you angles; a Banach space gives up the angles but keeps the most essential thing — a working notion of size that never lets sequences drift off the edge of the space. It is a normed vector space that is complete. You can measure ||v||, you can add and scale, and any sequence whose terms huddle arbitrarily close together (a Cauchy sequence) is guaranteed to converge to a genuine point of the space.

Precisely: a Banach space is a vector space X with a norm ||.|| satisfying ||v|| >= 0 with equality only at 0, ||c v|| = |c| ||v||, and the triangle inequality ||u + v|| <= ||u|| + ||v|| — and which is complete in the metric d(u,v) = ||u - v||. There is no inner product, so there is no <u,v>, no automatic notion of perpendicularity, and the parallelogram law need not hold. What survives is everything you can phrase using only the norm.

Why bother dropping the inner product? Because most spaces analysts care about have no natural inner product. The space C[a,b] of continuous functions with the sup-norm, the sequence spaces ell^p for p != 2, the Lebesgue spaces L^p for p != 2 — all Banach, none Hilbert. The big theorems of functional analysis (Hahn-Banach, open mapping, closed graph, uniform boundedness) live at exactly this level of generality, needing completeness but not orthogonality.

Relationship to Hilbert: every Hilbert space is a Banach space (forget the inner product, keep its norm), but not conversely. A Banach space is Hilbertian precisely when its norm satisfies the parallelogram law — that identity is the fingerprint that an inner product secretly generates the norm.