aThe Cauchy condition for uniform convergence. Theorem 13-4: Let {fn} be a sequence of functions defined on a set T. There exists a function f such that fn ® f uniformly on T if, and only if, the following condition (called the Cauchy condition) is satisfied: Cauchy条件为一致收敛。
We shall show that this net is Cauchy in the norm topology of X. Let any ε > 0 be given; let δ = δ(ε) be the modulus of convexity of the space (as in 22.40). By definition of the norm of X**, there is some f∈ X* with ||f|| = 1 and Re〈ξ, f〉 > 1 − ...
Passing in the limit as m,n\rightarrow +\infty implies \lim _{m,n}\Vert {\bar{x}}_n-{\bar{x}}_m\Vert =0. Hence, ({\bar{x}}_n)_{n \in {\mathbb {N}}} is a Cauchy sequence in {{\,\mathrm{Fix}\,}}T. Since {{\,\mathrm{Fix}\,}}T is a closed set and hence...