By Curtis F. Gerald Patrick O. Wheatley

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**Extra resources for Applied Numerical Analysis - Solutions manual**

**Example text**

1. Suppose that each of H and K is a Hilbert space and T is a closed and densely deﬁned linear transformation on H to K. Let H be the Hilbert space whose points are those of D(T ) where x H = (Txx ) H×K , x ∈ D(T ). 18) Suppose that the linear transformation T0 is a closed, densely deﬁned restriction of T (see [204] for a discussion of closed unbounded linear operators from one Hilbert space to another). Denote by H0 the Hilbert space whose points are those of D(T0 ) where x H0 = (Tx0 x ) H×K , x ∈ D(T0 ).

10) for every bounded subset Ω of H. 1) holds, or else (ii) lim z(t) t→∞ H = ∞. Proof. 10) holds for every bounded subset Ω of H. Suppose furthermore that R(z) is not bounded but nevertheless does not satisfy (ii) of the theorem. Then there are r, s > 0 ∞ so that 0 < s < r, and two unbounded increasing sequences {ri }∞ i=1 , {si }i=1 so that si < ri < si+1 , z(si ) H = s, z(ri ) s ≤ z(t) H H = r, ≤ r, t ∈ [si , ri ], i = 1, 2, . . 7, this is impossible and the theorem is established. A similar phenomenon has been indicated in [11] for semigroups related to monotone operators.

Q Pick p, q so that p ∈ (1, 2), p = 1−q , H is embedded in Lq (Ω) and, if f, g ∈ H, then f h ∈ Lp where h is any partial derivative of g. Deﬁne Sp = {f = (f (0), f (1), . . , f (n))} with n f Sp = f (0) Lp + f (i) K. i=1 If f ∈ Lp , and u ∈ H, use the notation f, u K to mean f u, Ω and deﬁne, for f ∈ Sp , u ∈ H, n f, u n+1 K to mean f (0)u + Ω f (i)ui , i=1 Ω where ui is the partial derivative of u in the i − th coordinate direction. Use w for some w ∈ H (after all, W is a collection u ∈ W to mean that u = ∇w of ordered pairs).