Lecture 3: Functional Analysis - revision of Metric and Topological Spaces

[@m.s.3588]

Thank you. around timestamp 32, why the distance between the two sequences equal to 1 and why there is no subsequence converges.

[@uniofnottingham]

Hi There, This is the response from Dr Feinstein: Here there is just one sequence of functions (not two). Each function f_n takes only values between 2 and 3. This forces |f_m(x)-f_n(x)| to be less than or equal to 1. However each function takes the value 3 at a point where all the other functions take the value 2. That forces the distance between the functions f_m and f_n to be exactly 1 when m is not equal to n. Suppose you have a sequence in a metric space, and the sequence has the property that the different terms of the sequence all have distance at least 1 from each other. Then that sequence cannot have a convergent subsequence. (Otherwise there would be two terms very close together, which would be a contradiction.)

[@mkminerals12343]

dear sir,we are watching your lectures,but we r the students of physics,please be in an easy way and its physical structures.thanks