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In some cases, the topology on V extends to some natural superspace W of more general functions that contain V ; in such cases, it is often 3This will be the only type of metric on function spaces encountered in this course. g. spaces of maps from one manifold to another) which have no vector space structure or norm, but still have a metric. g. 6 of Poincar´ e’s Legacies, Vol. I. 3. Lp spaces 29 important to know whether V is closed in W , so that limits of sequences in V stay in V . • Functional structures.

If µ is unsigned, then µac and µs are also. 9. If every point in X is measurable, we call a signed measure µ continuous if µ({x}) = 0 for all x. 6, but suppose also that every point is measurable and m is continuous. Show that there is a unique decomposition µ = µac + µsc + µpp , where µac m, µpp is supported on an at most countable set, and µsc is both singular with respect to m and continuous. Furthermore, if µ is unsigned, then µac , µsc , µpp are also. We call µsc and µpp the singular continuous and pure point components of µ respectively.

Yn ) := x1 y1 + . . 33) |x| = (x · x)1/2 . 34) x·x≥0 with equality if and only if x = 0. 36) x · y = y · x. These properties make the inner product easier to manipulate algebraically than the norm. 37) (z1 , . . , zn ) := |z1 |2 + . . 38) (z1 , . . , zn ) · (w1 , . . , wn ) := z1 w1 + . . 39) x = ( x, x )1/2 . 48 1. 40) x, x ≥ 0, with equality if and only if x = 0. 42) x, y = y, x in order to be compatible with sesquilinearity. We can formalise all these properties axiomatically as follows.

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