Banach space

A Banach space is a vector space equipped with a norm that is complete: every Cauchy sequence in its norm converges to an element of the space.

Normed vector space: A vector space equipped with a norm that assigns lengths to vectors and induces a notion of distance. Banach spaces are precisely the complete members of this broader class.
Banach fixed-point theorem: A theorem stating that a contraction on a nonempty complete metric space has a unique fixed point. Completeness ensures the successive approximations converge to the fixed point.
ℓp space: A sequence space of scalar sequences whose absolute values raised to a fixed power have finite sum, for p at least one. Each ℓp space is a standard Banach space under its usual norm.
Banach space-valued differential equation: A differential equation whose unknown function takes values in a Banach space. Completeness supports existence proofs for solutions through fixed-point methods.
Incomplete normed space: A normed vector space containing a Cauchy sequence that has no limit within the space. It fails exactly the completeness condition that distinguishes Banach spaces.
Norm: A function on a vector space that assigns nonnegative lengths and satisfies homogeneity and the triangle inequality. The norm defines the distance and convergence used in Banach-space completeness.
Banach–Steinhaus theorem: A theorem stating that a pointwise bounded family of continuous linear operators on a Banach space is uniformly bounded. The theorem turns pointwise control into a uniform bound using completeness.
ℓ∞: The space of bounded scalar sequences equipped with the supremum norm. Its supremum norm makes bounded sequences into a complete normed space.
Picard–Lindelöf theorem: An existence-and-uniqueness theorem for ordinary differential equations with a locally Lipschitz right-hand side. Its standard proof constructs solutions by contraction on a complete function space.
Pre-Hilbert space: An inner-product space that is not required to be complete in the induced norm. Completing a pre-Hilbert space yields a Hilbert space, illustrating the role of completion.

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