*Appendix B*

**SOLUTIONS TO EXERCISES**

**1.1.10**: Take approximating sequences {*ƒ*_{n}}, {*g*_{n}} ⊂ for *ƒ*, *g*. On account of the proof of Theorem 1.1.5(ii), we may assume that |*ƒ*_{n}| ≤ ||*ƒ*||_{∞}, |*g*_{n}| ≤ ||*g*||_{∞}. Since

|*ƒ*_{n}(*x*)*g*_{n}(*x*) – *ƒ*_{n}(*y*)*g*_{n}(*y*)| ≤ ||*g*||_{∞}|*ƒ*_{n}(*x*) – *ƒ*_{n}(*y*)| + ||*ƒ*||_{∞}|*g*_{n}(*x*) – *g*_{n}(*y*)|,

and |*ƒ*_{n}(*x*)*g*_{n}(*x*)| ≤ ||*g*||_{∞}|*ƒ*_{n}(*x*)| + ||*ƒ*||_{∞}|*g*_{n}(*x*)|, we get from (1.1.13) that *A*_{T}_{t}(*ƒ*_{n}g_{n})^{½}≤ ||*g*||_{∞}*A*_{Tt}(*ƒ*_{n})^{½} + ||*ƒ*||_{∞}*A*_{Tt}(*g*_{n})^{½}. Dividing by *t* and letting *t* ↓ 0, we have ||*ƒ*_{n}g_{n}||_{E} ≤ ||*g*||_{∞} ||*ƒ*_{n}||_{E} + ||*ƒ*||_{∞} ||*g*_{n}||_{E}. Since ||*ƒ*_{n}g_{n}||_{E} is uniformly bounded and *ƒ*_{n}g_{n} converges to *fg* as *n* → ∞, ...