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lR such that ¢I XO = ¢o and ¢(x) :::; p(x) Vx E X.

4. 2 (1) Where is the completeness of H used in the proof of the Riesz lemma; more precisely, what can you say about X* if you only know that X is a (not necessarily complete) inner product space? ) "'I (2) 1fT E £(H,K), where H,K are Hilbert spaces, prove that IITII = sup{I(Tx,y)l: x Ilxll :s: E H,y E K, 1, Ilyll :s: I} . (3) If T E £(H) and if (Tx, x) = 0 V x E H, show that T = O. 5(2). Notice that this proof relies on the fact that the underlying field of scalars is C - since the Polarisation identity does.

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