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Let V : L2{ii\) —> L2(fj,2) be a linear isometry, A C L°°(/i 1 ) be an additive group and g 6 L2(f/,i). 2) is true for every f G A, the closure of A in L°°(fii). Proof. Let f & A and choose a sequence {/„} Q A so that ||/„ - /j|oa -> °- T h e n Wfn9-f9h - > 0 a n d hence | | V ( / „ s ) - V ( / f f ) | | z -> 0 because V is an isometry. 2) and the choice of {/„}, it follows that \\Vfn ~ Vf\\oo -» 0. Thus \\Vfn -Vg-VfVg\\2 -» 0. Now since V(fn9) = Vfn ■ Vg, ||V/„||co = l l / n | U n>l, we have by letting n -> oo that V(/ff) = V / ■ Vp and ||V/||oo = ll/lloo as desired.

3) (3) 7/ ||/i n - un\\ -► 0 and \\vn - v\\ -» 0 witfi {(i„,/i, i/„, v : n > 1} C P{X), then H(n\u) < liminf Hfa\un). 4) n—+oo (4) 7/ fi,veP(X), then \\n-V\\

D / / G M S ( X ) . Moreover, it is linear. The boundedness of H(-,% S) follows from |#(£,2l,S)| =