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8 Oct 2021: 3. (i) Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive definite bilinear form on the spaceMatn(R) of all nn real matrices. ... sn forms a basis for Pn.(iii) For all 1 k n, sk spans the orthogonal complement of Pk1 in Pk.(iv) sk is an

29 Nov 2021: set is a union of somemembers of B; equivalently, for U X, we have that U is open in X if and only iffor all x U there exists B B ... Show that if fn Y for all n N andfn f in X, then f Y.

12 Feb 2021: every z C, |f(z) b| > ε.(iii) f = u iv and |u(z)| > |v(z)| for all z C. ... z|)k for all z.(ii) Show that an entire function f is a polynomial of positive degree if and only if |f(z)|as |z|.

8 Aug 2021: Correction from Greg Price Last line of Exercise 11.5.25 should read:-Show that T̂ s′ and T̂ 6= 0, but T̂b = 0 for all b c00. ... least n 1 diagonalelements having value 1 and all diagonal elements non-zero.’.

11 Feb 2021: thetransition functions for the atlas with charts {π+,π,φ} are all smooth. ... b) Prove that for any decomposition of S (where all vertices have valence at least 3), we have 2E/F 6 (1 χ(S)/F),where χ(S) is the Euler

15 Oct 2021: a) f L1(Rn), g Ck(Rn) with supRn |Dαg| < for all |α| k. ... 3. for all j J. Thuslimj. τzjg gLp(Rn) = 0.4This is another point at which p 6= is crucial.

20 Apr 2021: The étale homotopy type of Spec k is "-isomorphicto the pro-object {B(Gal(K/k))} where K/k runs over the system of all finite Galois extensions K/k. ... So Et! (Spec k) is isomorphic to the pro-object {B(Gal(K/k))}.This implies that all the higher

16 Mar 2021: Show that there exists aconstant Cn, > 0 such that for all x, y 2 Rn:. ... u |x|2u = f (†). if. (u, v)H = (f, v)L2 for all v 2 H. ().

15 Oct 2021: ii) For all f C0([a,b]), if a R, then ||af||C0 = |a| ||f||C0. ... fi(x)fi(y)| < ,for all i N and all x,y [a,b] with |xy| < δ.

15 Oct 2021: f ′(x) = g′(x), for all x (a,b). Show that:f(x) = g(x) C. ... f(k)(x) = g(k)(x), for all x (a,b). Show that:f(x) = g(x) pk1(x).