AYTUNA - KRONE - TERZiOÔLU

m - l

\p'nLk { y ) - p'mLmiy)^':^ < E \pHiLi+iiy) - р^ыу)^':'^

m - 1

<IlCf\fj + , Lj + r { y ) - Lj { y ) \^^

m—1 -

<Т>ФМ\о< : ф^\\у\\о .

Hence the limit of {Рт^т{у))т=п exists in Xl*^\ which we denote by S^^\y). Further from

| / , * L „ ( y ) - 5< * ) ( y ) |f , <2^IM|o

we see that 5^*) : Y -♦ Xj^**^ is continuous. For m> к and у € X^"*^ we have

/ '^5 ( ' " ) ( î / ) = lim ^>^;.r ^/(У) = lim р^Ну) = 5(*)(у).

Here we are considering each p*j as a bounded map of Xm^ into X^^^K Since E can be represented as the projective limit of {Xj^^^;p!^) there is a continuous linear map L :Y -^ E with phL^ S^^\ where pk '- E —^ ^* is the canonical inclusion.

If L(y) = 0, since S^^\y) = p\L{y) = 0 we have

^ < Ibi(y)ir) = \L.{y) - 5(^)(,)|p) < 2щ||у||о

which gives у = 0. So L is one-to-one.

Suppose MmЬ{ут) = я?о. If (||Ут||о) is bounded, from

|Ь * ( Ут ) - 5 ( * ) ( у „ . ) й ) <2щ||г / т||о

we obtain

1

Ут\\к

< \Lk{ym)t^ < ^Ibmllo + \PkL{ym)&

and so we see that (ут) is a bounded sequence in Y. Hence it has a subsequence which converges to some у and therefore Ly = xq,

If (||2/m||o) is not bounded, by passing to a subsequence we may assume Ibmllo Î 00 and set Zm = ЦУтЦо^Ут. Then limL(2;rn) = 0 and ||zr„||o = Ь By the above argument we have a z G У which is the Umit of a subsequence of (zm) and so L{z) = 0 but ||z||o = 1. This contradicts the fact that L is one-to-one. Hence L has a closed range and thus it is an imbedding of У into E, D

Remarks . For e stable Лоо(е) is isomorphic to a subspace of Ai{e) ([7]) and any subspace У of Ai{e) which has property (DN) satisfies Лоо{еУ С A{Y)

139