By A. I Markushevich

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Vi. c V). 8) are satisfied, and hence Theorem 2 is applicable. •. , Wn (cf. 2), and so Theorem 2 also contains Theorem 1 of Chapter 2. 2 is the following. /e, an injective linear operator Ph from Vh into V and a linear operator 'h from V into Vh' We suppose that 38 NUMERICAL ANALYSIS (i) Phrh is a continuous linear operator in V; (ii) for all VE V: lim Phrhv = v in the strong topology of V; h .... O (iii) the norms of the operators Phrh have an upper bound independent of h. 2 We provide the space Vh with a scalar product: and the Vh become Hilbert spaces.

Stable and convergent Hilbert approximation be given of the space V, say {Vh' Ph' rhhe;t'. 2) IIlhll*h~P, 32 NUMERICAL ANALYSIS in which II . II *h stands for the norm of V~ and in which Pis independent of h. 2). Remark. 3) can be written (cf. 2. Convergence Theorems We want to study the convergence of Uh to U, so we must first make precise the way in which the forms ah and Ih approximate the forms a and I. 5) If the family Vh converges weakly to v for h -+ 0, then, for all WE V lim ah(vh, rhw) = a(v, w).

2 This follows in fact from (i) and (ii) by the theorem of BanachSteinhaus. 3 Numbers in square brackets refer to the Bibliography on p. 161. 1 CHAPTER 5 A METHOD FOR COMPUTA TION OF THE ERROR We go back to the situation of Chapter 4 and more precisely to the framework of Theorem 2, and we want to estimate the error between u and Uh' This is not possible merely with the hypotheses of Chapter 4, and we are only going to indicate some methods for error computation, which subsequently must be developed in the examples.

### Areas and logarithms by A. I Markushevich

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