0 for each A E K.

F, then, clearly, G = G1 n G2 , where G1 = {x: g(x) = O(}, G2 = {x: g(ix) = -f3}. Since f is determined by C '0 within a non-zero factor, G1 and G2 determine the unique pencil whose intersection is G. 2), G1 and G2 are closed if and only if G is closed in L. If L is a vector space over a field HeR, there does not always exist an automorphism u of L satisfying u2 = -e; examples are furnished by real vector spaces of finite odd dimension. It is still often desirable, especially for the purposes of spectral theory, to imbed L isomorphically into a vector space over K = H(i); the following procedure will provide such an imbedding.

We show that p = PM' It follows from (b) that {x: p(x) < A} = AM for every A> 0; hence if p(x) = rx, then x E AM for all A> rx but for no), < rx, which proves thatp(x) = inf {A> 0: x E AM} = PM(X). 4) and will be omitted. 5 Let M be a radial, convex, circled subset of L;for the semi-norm p on L to be the gauge of M, it is necessary and sufficient that Mo c Me M 1 , where Mo = {x: p(x) < I} and M] = {x: p(x) ;2; I}. If L is a topological vector space, the continuity of a semi-norm p on L is governed by the following relationship.

### Archimedes' Revenge by Paul Hoffman

by Edward

4.3