77.8. ADDING A QUASI-BOUNDED OPERATOR 2645

Then if m,n > M,

|⟨vn− vm,un−um⟩|= |⟨vn,un⟩+ ⟨vm,um⟩−⟨vn,um⟩−⟨vm,un⟩|< ε

Hence it is also true that

|⟨vn,un⟩+ ⟨vm,um⟩−⟨vn,um⟩−⟨vm,un⟩| ≤ |2µ− (⟨vn,um⟩+ ⟨vm,un⟩)|< 3ε

Now take a limit first with respect to n and then with respect to m to obtain

|2µ− (⟨v,u⟩+ ⟨v,u⟩)|< 3ε

Since ε is arbitrary, µ = ⟨v,u⟩ after all. Hence the claim that ⟨vn,um⟩ → ⟨v,u⟩ is verified.Next suppose [x,y] ∈ G (G) and consider

⟨v− y,u− x⟩= ⟨v,u⟩−⟨v,x⟩−⟨y,u⟩+ ⟨y,x⟩

= limn→∞

(⟨vn,un⟩−⟨vn,x⟩−⟨y,un⟩+ ⟨y,x⟩)

= limn→∞⟨vn− y,un− x⟩ ≥ 0

and since [x,y] is arbitrary, it follows that v ∈ Gu.Next suppose limsupn→∞ ⟨vn− v,un−u⟩ ≤ 0. It is not known that [u,v] ∈ G (G).

lim supn→∞

[⟨vn,un⟩−⟨v,un⟩−⟨vn,u⟩+ ⟨v,u⟩] ≤ 0

lim supn→∞

⟨vn,un⟩−⟨v,u⟩ ≤ 0

Thus limsupn→∞ ⟨vn,un⟩ ≤ ⟨v,u⟩. Now let [x,y] ∈ G (G)

⟨v− y,u− x⟩= ⟨v,u⟩−⟨v,x⟩−⟨y,u⟩+ ⟨y,x⟩

≥ lim supn→∞

[⟨vn,un⟩−⟨vn,x⟩−⟨y,un⟩+ ⟨y,x⟩]

≥ lim infn→∞

[⟨vn− y,un− x⟩]≥ 0

Hence [u,v] ∈ G (G). Now

lim supn→∞

⟨vn− v,un−u⟩ ≤ 0≤ lim infn→∞⟨vn− v,un−u⟩

the second coming from monotonicity and the fact that v ∈ Gu. Therefore,

limn→∞⟨vn− v,un−u⟩= 0

which shows that limn→∞ ⟨vn,un⟩= ⟨v,u⟩.Similar reasoning implies