# A0-stable linear multistep formulas of the-type by Rockswold G. K. By Rockswold G. K.

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7). 30 to find the unique solution for the Cauchy problem of a linear differential equation. 3. Hilbert spaces Let us review some basic facts concerning Hilbert spaces. 1. Scalar products. A scalar product or inner product in a real or complex vector space H is a K-valued function on H x H, (x,y) E H x H H (x,y)H E K, having the following properties: (1) It is a sesquilinear form, meaning that, for every x E H, is a linear form on H and (x, )H is skewlinear, that is, x)H (x,yi + y2)x = (x,y1)H+(x,y2)H, (X,Ay)H = (x,y)H.

Suppose that f : X -+ Y, where X and Y are two topological spaces, and xo e X. (a) If X is metrizable (or if xo has a countable neighborhood basis), prove that f is continuous at xp if and only if f is sequentially continuous at xp. (b) Let X be R endowed with the topology of all sets G C R such that GC is countable, and let Y also be R but with the discrete topology (all subsets of R are open sets). Show that Id : X --+ Y is a sequentially continuous noncontinuous function. 10. If f is a measurable function on a measurable space, show that Sgn f(x) := f fix) (with 0/0 := 0) defines another measurable function.

E. 10. LP() is a Banach space. Proof. Assume first 1 < p < oo and let {fk} be a Cauchy sequence in LP(µ). As in the preceding proof of (c), there exists a subsequence {fin} which is convergent to a function h E LP(). By the triangle inequality, we also obtain fk - h in LP(). In L°°(µ), if {fk} is a Cauchy sequence, the sets Bk :_ {x; I> Mfklloo} and Bp,9 :_ {x; Ifp(x) - f9(x)I> 'lfp have measure 0, and also µ(B) = 0 if B is the union of all of them. Then we have limk fk(x) _ 1(x) uniformly on B° and limy fk =fin L°°(µ).

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