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Banach Spaces (Definition)

Complete normed vector spaces, central to functional analysis.
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The statement of the theorem

Let K\mathbb{K} be the field of scalars, R\mathbb{R} or C\mathbb{C}. A Banach space is a pair (V,∣∣⋅∣∣)(V, ||\cdot||), where VV is a vector space over K\mathbb{K}, and ∣⋅∣∣|\cdot|| is a norm on VV, such that the induced metric d(x,y)=∣∣x−y∣∣d(x, y) = ||x - y|| makes VV a complete metric space. \n\nFormally, the norm ∣⋅∣∣:V→R|\cdot||: V \to \mathbb{R} must satisfy the following axioms for all x,y∈Vx, y \in V and α∈K\alpha \in \mathbb{K}:\n\n1. Non-negativity and Definiteness: ∣∣x∣∣≥0||x|| \ge 0, and ∣∣x∣∣=0  ⟺  x=0||x|| = 0 \iff x = 0.\n2. Homogeneity: ∣αx∣∣=∣α∣⋅∣∣x∣∣|\alpha x|| = |\alpha| \cdot ||x||.\n3. Triangle Inequality: ∣∣x+y∣∣≤∣∣x∣∣+∣∣y∣∣||x + y|| \le ||x|| + ||y||.\n\nFurthermore, VV must be complete, meaning that every Cauchy sequence in VV converges to an element in VV. That is, for any sequence (xn)n=1∞⊂V(x_n)_{n=1}^{\infty} \subset V, if for every ϵ>0\epsilon > 0, there exists an integer N∈NN \in \mathbb{N} such that for all m,n≥Nm, n \ge N, we have ∣∣xm−xn∣∣<ϵ||x_m - x_n|| < \epsilon, then there exists an element x∈Vx \in V such that lim⁡n→∞∣∣xn−x∣∣=0\lim_{n \to \infty} ||x_n - x|| = 0.