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A sequence {x_n} in a Banach space converges to x if ||x_n - x|| → 0 as n → ∞.

A Banach space is separable if it contains a countable, dense subset.

A linear operator T between normed spaces is bounded if there exists a constant C such that for all x, ||T(x)|| ≤ C·||x||.

A sequence {x_n} in a Banach space X weakly converges to x if for every continuous linear functional f in X*, f(x_n) → f(x) as n → ∞.

The Hahn-Banach Theorem extends a bounded linear functional defined on a subspace to the whole space without increasing its norm.

The dual space of a normed space X, denoted by X*, is the set of all bounded linear functionals on X.

A Banach space is reflexive if it is isometrically isomorphic to its double dual, i.e., X ≅ X**.

The Banach-Alaoglu Theorem states that the closed unit ball in the dual space of a normed space is compact in the weak* topology.

A Schauder basis is a sequence {x_n} in a Banach space such that every element x in the space can be written as a convergent series x = Σ α_n x_n.

An Lp space is a vector space of measurable functions for which the p-th power of the absolute value is Lebesgue integrable.

A Banach space is a vector space X with a norm ||x|| such that X is complete with respect to the metric induced by the norm.

A metric space is complete if every Cauchy sequence in the space converges to an element within the space.

A sequence {x_n} in a normed space is Cauchy if for every ε > 0, there exists an N such that for all m, n ≥ N, ||x_m - x_n|| < ε.

The Uniform Boundedness Principle states that for a family of bounded linear operators, if each point is taken to a bounded set, then the operators are uniformly bounded.

The Open Mapping Theorem states that if T is a continuous, surjective linear operator from one Banach space onto another, then T maps open sets to open sets.

The space C([a,b]) of real-valued continuous functions on a closed interval [a,b] with the supremum norm ||f||_∞ = sup{|f(x)| : x ∈ [a,b]} is a Banach space.

A norm on a vector space X is a function ||x||: X -> [0,∞) that satisfies positivity, homogeneity, and triangle inequality.

Every contraction mapping on a non-empty complete metric space has a unique fixed point.

A linear operator is a function between two vector spaces that preserves vector addition and scalar multiplication.