Question 1.
1) Suppose (V, | * |) is a normed space. If x, y E V and r is a positive real number, show that the open r-balls Br(x) and Br(x + y) in V are homeomorphic.
2) Suppose V and W are two normed spaces. If A : V ---> W is a linear map, then show that it is continuous at every point v E V if and only if it is continuous at 0 E V.
3) Suppose A: (V, | * |) ---> (W, | * |w) is a linear map between normed spaces, and there is a number R E R such that |A(v)|w <= R|v|_v for all v E V. Explain why A is continuous.
Question 2
Let (0,1) denote the open unit interval in R, and C(0,1) the set of all continuous functions (0,1) ---> R. Is C(0,1) a subset of B((0,1),R) the set of all bounded functions on (0,1)? Is C(0,1) a normed space with the sup-norm | * | given by |f| = sup_(t E (0,1){|f(t)|}?
Question 3
1) For a compact topological space, and Y a compacct subset of X. The inclusion i: Y --->X gives a function i* : B(X,R) ---> B(Y,R), from the bounded functions on X to the bounded functions on Y by i*(f) = f in i for each f E B(X,R). Explain why i* is surjective.
2) Using the sup norm | * | on both these sets of bounded functions, for a function f E B(X,R), what, is any, is the relation between |f| and |i* (f)|. Is i* continuous?
3) For a compact topological space X, denoted by C(X) the banach space of continuous functions on X with the usual sup-norm.Following the idea of part (1), explain why a continuous function alpha: X --->Z between two compact topological spaces gives a function alpha*:C(Z) ---> C(X)
4) Explain why alpha* of part (3) is a linear map
5) Explain why alpha* of part (3) is continuous.
6) If I: X ---> X is the identity function, show that I*, as in part (3), is the identity function C(X) ---> C(X).
7) If alpha: X_1 ---> X_2 and beta: X_2 ---> X_3 are continuous functions of compact topological spaces, explain why (beta is in alpha)* = alpha* is in beta*
8) Hence prove that if gama: X --> Z is a homeomorphism of compact topological spaces, gama*: C(Z) ---> C(X) is a homeomorphism.