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- Q3) Let L = Z* and |x ||= max {|x1 - x3l, x2l, 15x,1} Vx = (x1, X2, X3, X4) E Z*. Is (Z*, ) is a normed space?Q/Let X+6 and (Y₁Ty) topological space and het fox-cy function. if Tx is defined as [x = {f(G) | Ge Ty} then (Txofological Space(2) Let (X,r)be a topological space, If for every A open set, and x E A, 3 Ux and Vy open sets, such that x E Ux, AC Vy, and Uxn Vy = 4. The space is called: (a) Hausdorff Space (b) Regular space (c) T1-space (d) Normal space
- Exercise 3. Are the following subsets of the real line with its usual topology (in which U CR is open if and only if for every x ‹ U there is € > 0 with (x − €, x + € ) ≤ U) open, closed, both or neither? Explain. (1) [1,3); (2) (1,3) U (5,00); (3) (-∞0, ∞0); (4) { neN} U {0}; (5) The set Q of rationals. .Show that (C(X),|| · ||∞,x) is a metric space.17. Show that if X is a normed linear space and if X is separable, then so is X/M.
- Prove that (Co0, 11·11,) is not a Banach Space for any 11. Check that the following functions on R² are norms: 1/p a) | (1,72) ||,= (1z1P" + \#a!P), 1spRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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