Fréchet Differentiability of the Norm in Operator Spaces 531

If we choose x„eDn{x*>Qo(x*)-i/n\\x*\\} we have

C\\X * \\ < -Q,ÀX* + X*) + Qoix*) + X*iXo)

й - { х * + X*) (xj+e,Àx*)+X„*(Xo)

^x * ( xo - x „ ) + ( l / n ) ||x „ * || ^l|x*||||xo-xJ|+(l/n)||x„*|| u\\x*Ub + i/n)

which is impossible. Thus we may assume that for all и

Qoix * + X*) - Qd{x*) - x*(xo) > с II x? II. Choose

у „ боп { ( х * + х„*)>ев(х* + х„*)-1/п||х*||}. Then we have

x * W= ( x * + x*)();„)-(x*)();„)

èMx * + ^n*)-i|x*||(l/n+||yJ|)--^^(x*).

For n large enough we have that ||y„-Xoi| ^b. It follows that

4xt\\<QD { x * + X * ) - Qj ) { x * ) - X^iXo )

^ ( x * + x „ * ) ( ) ; J + (l/n)||x*||-^^*)-x*(xo)^(l/n + b)||x„*L

Again this is a contradiction.

Step 2, Suppose that we have x* in X*, a>0, and b>0 so that

<5д ( х * , ä)Sb<2b< ад(х*, à).

Choose x^* in the а{Х**,Х*) closure of D in Z** so that öd(^*) = x5*(x*). By assumption there exists x in Dn{x*>Qj){x*)-a} such that ||xj*-x|| >b. Choose ^1 in Sx* so that

( xS * - x ) ( x ? ) >b .

Then for any oO we have

с ~ ^{Qo{x* + с • XÎ) - Qoix*) - с • x?(x))

è C" \xt4x* + с • Xf) - Qjyix*) - с • Xf{x))

= xf{x^* — x)>b. This contradicts the assumption that (5д(х*,а)^Ь. ^'•00/of 2.2 (b). Everything follows from (a) except that

a^ ( x * ) ^3№ * ) .