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Here are some consequences of this assumption. 1. Assuming the axiom of choice, every vector space V over F has an algebraic basis (that is to say a subset E such that any v ∈ V may be written uniquely as a finite sum v= λg g g∈G with G a finite subset of E and λg ∈ F). 12 For historical reasons this axiom has acquired an air of glamour and mystery which it it hardly deserves. 2. 2. Assuming the axiom of choice, if U is any infinite dimensional normed vector space over F, then there exists a linear map α : U → F which is not continuous.

Show that Tn ∈ L(l1 , c00 ). Show that the Tn form a Cauchy sequence in L(l1 , c00 ) with no limit point. 6 may fail if (V, . ) is not complete. 9. If T : U → V is an isomorphism between the Banach spaces U and V (that is to say, a linear bijection such that T and T −1 are continuous), show that the map T ′ : V ′ → U ′ between the dual spaces given by T ′ (v ′ )u = v ′ (T u) for all v ′ ∈ V ′ and u ∈ U is a well defined isomorphism between V ′ and U ′ . ) Deduce that l1 cannot be isomorphic to lp for any p > 1.

Ii) Show, by means of an example, that the condition (X, d) compact cannot be dropped. Show, by means of an example, that the condition fn decreasing cannot be dropped. Show, by means of an example, that the condition fn continuous cannot be dropped. (iii) Set p0 = 0 and pn+1 (x) = 12 x2 + pn (x) − 12 pn (x)2 . Explain why pn is a polynomial. Show that pn (x) ≤ pn+1 (x) ≤ |x| and all n ≥ 0 for all x ∈ [0, 1]. Hence deduce that pn (x) → |x| as n → ∞ for all x ∈ [0, 1]. Now use Dini’s theorem to show that the convergence is uniform.

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