Characterization of M(H, K)X

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                                          4.2 Related convergence results   63

  (M (Hn , Hn )X  M (H, H)X) 

      M (Hn Pn , Hn Pn )(Pn XPn )  M (HP  , HP  )(P  XP  )  
            (2  4L(, ))X  .

Therefore, we can repeat the arguments at the end of the part (c) in the proof of Theorem 4.1 to get the desired convergence; in fact, use the above estimate in place of (4.5) and apply Proposition 4.3 (see also Remark 4.2, (1)) together with Pn XPn P XP 0, which is a consequence of the compactness of X. We point out that the arguments in Case (b) in Theorem 4.1 gives us the norm convergence M (Hn , Kn )X M (H,...

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