Lectures on Functional Analysis
نویسندگان
چکیده
The corresponding topology is called norm-topology. The norm-topology naturally gives rise to the concept of convergence and continuity. A sequence txnun“1 Ă X is said to converge to x in the norm-metric, or strongly, if }xn ́ x} Ñ 0 as nÑ 8. A function f : X Ñ Y , where Y is a topological space, is continuous if fpxnq Ñ fpxq whenever xn Ñ x, or equivalently, if f ́1pUq is open for any open U Ă Y . Exercise 1.2. Prove that the norm } ̈ } : X Ñ R is a continuous function on X. A series 8 n“1 xn is said to converge to its sum x P X if the sequence of partial sums řN n“1 xn tends to x. A series converges absolutely if the numerical series 8 n“1 }xn} converges. The natural fact that an absolutely convergent series itself converges does not necessarily hold in general normed spaces. To ensure this property one has to impose an additional completeness assumption. A metric space pM,dq is complete if its every Cauchy sequence has a limit, i.e. if dpxn, xmq Ñ 0 implies Dx PM such that dpxn, xq Ñ 0. A complete normed space pX, } ̈ }q is called a Banach space.
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تاریخ انتشار 2013