Question

Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in class is a valid norm on R2 (b) Prove that in R2, any open ball in 12 ("Euclidean metric") can be enclosed in an open ball in the loo norm ("sup" norm). (c). Say I have a collection of functions f:I R. Say I (1,2). Consider the convergence of a sequence of functions fn (z) → f(x) in 12-Show that the convergence amounts to showing that fn(zi)-rx.) for a fixed xi, i=1,2 (d) Say now 1 [0,1], and fn(x) : 1-+ R for each n, and each fa(z) İs a continuous function on I. Let fn(x) f(x) for each point r [0, 1]. Will f(x) be a continuous function? If so, prove it. If not, given an example of the failure of this convergence to a continuous function (and explain the problem). 2. Show that any normed vector space is a metric space, but the converse is not true. 3.1 Consider a function f: R" R. Let the domain for the function have either the "sup" norm/metric or the Euclidean norm/metric on R". Explain why if f is continuous from Rn to R in the Euclidean metric its also contin- uous when using the "sup" norm/metric on R". (hint: follow my discussion in the slides...use the definition of continuity of a function, and then the rela- tionship between open sets in the Euclidean metric and open sets in the "sup" norm/metric.)

Answer 1

Every Normal vector space/line space is a metric space. But converse is not true. Let us define d: N times N rightarrow R (i) As d (x, y) greaterthanorequalto 0 Forall x, y as always positive. (ii) d (x, y) = Vertical-bar Vertical-bar x - y Vertical-bar Vertical-bar = Vertical-bar Vertical-bar -(y - x) Vertical-bar Vertical-bar = Vertical-bar Vertical-bar y - x Vertical-bar Vertical-bar = d (x, y). (iii) d (x, y) = Vertical-bar Vertical-bar x - y Vertical-bar Vertical-bar d (x, z) = Vertical-bar Vertical-bar x - z Vertical-bar Vertical-bar = Vertical-bar Vertical-bar x + y - y - z Vertical-bar Vertical-bar = Vertical-bar Vertical-bar x - y + (y - z) Vertical-bar Vertical-bar = Vertical-bar Vertical-bar x - y Vertical-bar Vertical-bar + Vertical-bar Vertical-bar y - z Vertical-bar Vertical-bar = d (x, y) + d(y, z) Therefore d (n, z) lessthanorequalto d (x, y) + d/y, (z) Therefore Every Line Space is Metric space Conversely: Let us define metric space d(x, y) = Vertical-bar Vertical-bar x, y Vertical-bar Vertical-bar d (x, y) = {0 1 x = y x notequalto y Let d(1, 9) = Vertical-bar Vertical-bar x - y Vertical-bar Vertical-bar = Vertical-bar Vertical-bar 11 - 9 Vertical-bar Vertical-bar = 2