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15 Oct 2021: αn! The spaces Ck() and C() are vector spaces over C, where addition and scalarmultiplication are defined pointwise. ... 1.1). Exercise(). Show that with the definitions (1.1) the space Ck() is a vectorspace over C, and that Cl() is a vector subspace of

6 Aug 2021: αn! The spaces Ck() and C() are vector spaces over C, where addition and scalarmultiplication are defined pointwise. ... 1.1). Exercise(). Show that with the definitions (1.1) the space Ck() is a vectorspace over C, and that Cl() is a vector subspace of

15 Oct 2021: Exercise 2.1 (). Suppose that we work over Rn and that f,g,h S. ... For any topological vector space V over C one candefine in a natural way the continuous dual space which consists of continuous linearmaps.

15 Oct 2021: You will have to think hard,and you may have to come back to some of the problems over more than one session tocrack them. ... x y :=(x1 y1,. ,xn yn. )t. If λ R, we define:λx :=. (λx1,. ,λxn. )t. With these definitions, Rn has the structure of a

15 Oct 2021: Then they remain close for all times t (T,T). We can improve the control over the solution to control higher derivatives:. ... are summed over 0,. , 3. Inorder to write the wave equation concisely, we introduce a (0, 2)tensor with componentsηµν called