Ques-1-Metric-Space-Question-Let-X-be-the-set-of-all-bounded-sequences-of-complex-numbers-That-is-every-element-of-is-a-complex-sequence-x-x-k-1-such-x-i-lt

Question Number 191786 by Mastermind last updated on 30/Apr/23

Ques . 1 (Metric Space Question)

Let X = ρ_{\infty} be the set of all

bounded sequences of complex

numbers . That is every element of

ρ_{\infty} is a complex sequence \bar{x} = {\bar{x}}_{k = 1}^{\infty}

such ∣ x_{i} ∣< K \bar{x}, i = 1, 2, 3, \dots where Kx

is a real number which may define

on x for an arbitrary \bar{x} = {x_{i}}_{i = 1}^{\infty} and

\bar{y} = {y_{i}}_{i = 1}^{\infty} in ρ_{\infty} we define as

d_{\infty} (x, y) = Sup ∣ x_{i} - y_{i} ∣, Verify that

d_{\infty} is a metric on ρ_{\infty .}