By Blok W.J.

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**Example text**

DEFINITION. A linear functional / on a normed linear vector space is said to be bounded if there is a constant c so that The smallest admissible constant in this inequality is called the norm of / and is denoted by ||/||*. Note that for two linear functionals lt and / 2 , and hence, Clearly ||a/||* = |a| ||/||*, and ||/||* = 0 implies that /(u) = 0 for all w, so that / is the zero functional. Thus the space V* of bounded linear functionals on the normed linear vector space V is again a normed linear vector space.

1. (a) (v,w) = 0/or all v in V implies w = 0. (b) The sesquilinear functional is Hermitian, and J^(M, u) is positive definite on V. 2 of Chap. 2). 1). 1). 2 of Chap. 2. 1. 1, the value is attained for some v = u t in V^. 2). Proof. 1 are satisfied. 1 (a) above/ ^ 0 implies u^ > 0. The linear condition (v,f) = 0 makes &(v, v) = 0. 5 there exists a ul in V^ with j/(u t , Mj = 1, ^(M! , u x ) = ul. 2), which completes the proof. We observe that /^ is the square of the norm of the linear functional (v,f).

1. 4) applied to £%(u, u) and —&(u, u) gives two sequences of eigenvalues Hi ^ u2 ^ • • • and —u\~) ^ — u(2~} ^ • • • , both of which converge to zero. 2) holds when convergence is defined in the norm stf(u, u) 1/2 and D is a solution of TV = 0. 4) are also valid. Remark. /(u, u). However, this sort of result is not valid if the domain D is allowed to become unbounded. Consider, for example, a domain D in 2-space which contains the infinite strip |x2| ^ a. Let (x,,x 2 ) be any infinitely differentiable function which vanishes outside the disk x] + x\ ^ a2.