By Korner T.W.
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Extra resources for Linear analysis
Sample text
20 Third example sheet Students who are unsure of their ground should check that they can do the exercises in the main text. Strong students should at least glance at the supplementary example sheet. The order of the exercises roughly follows the order of the lectures. 1. We work with the set A(D) of functions analytic on D = {z ∈ C : |z| < 1}. (i) Let 1 > R > r. By considering Cauchy’s formula with the contour z = Reiθ , or otherwise, show that FR (M ) = {f ∈ A(D) : |f (z)| ≤ M for all |z| ≤ R} is equicontinuous at every point z with |z| ≤ r.
2 depend on clever inequalities13 but there are other ways of arriving at the results. Let ∞ ≥ s > r ≥ 1. Investigate maxima and minima of nj=1 xsj subject to xj ≥ 0, nj=1 xrj = 1 using the calculus of variations. 4. If V is a vector space over F, we say that E is an algebraic basis (that is to say a basis in the sense of 1B algebra) if every v ∈ V can be written uniquely as a finite sum n v= λj ej j=1 with e1 , e2 , . . , en distinct elements of E and λj ∈ F. The collection V ∗ of linear maps α : V → F is called the algebraic dual (that is to say, the dual space in the sense of 1B algebra).
In this question you may use results about Riemann integration in one dimension but not in higher dimensions. (i) Suppose that f : [0, 1]2 → R is continuous. Explain why the function g : [0, 1] → R defined by 1 f (x, y) dy g(x) = 0 is continuous and so 1 1 f (x, y) dy 0 0 exists. 46 dx (ii) Show that, if u, v : [0, 1] → R are continuous, 1 1 1 u(x)v(y) dy 0 1 dx = u(x)v(y) dx 0 0 dy. 0 (iii) Use the theorem of Stone–Weierstrass to show that the collection of functions h of the form n h(x, y) = uj (x)vj (y) j=1 with uj , vj : [0, 1] → R continuous, is uniformly dense in C([0, 1]2 ).
Linear analysis by Korner T.W.
by Brian
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