By Cohn P.M.

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0 It is useful to know when such natural functionals are continuous. 15 Definition. Given a normed linear space (X, 11·11), a Schauder basis {en} is said to be monotone if for each x = n I. Akek the sequence {II I. Akek II} is monotone ~I ~I increasing. In most classical sequence spaces the obvious Schauder bases are monotone. For monotone bases we have the following satisfying property. 16 Theorem. Given a normed linear space (X, 11·11), with a monotone Schauder basis {en·), the corresponding coordinate functionals are continuous.

An attempt to overcome this problem by enlarging the space raises other difficulties. (ii) We could consider "0\[a,b], the linear space of all Riemann integrable functions on the interval [a,b]. In this case b (f, g) =f f(t) g(t) dt a is a positive Hermitian form but it is not positive definite. In the real linear space "01. [0, I] the function f defined by f(t) = 0 o::;t

3. Consider the linear space ~(X) of bounded functions on a non empty set X. Prove that ~(X) is finite dimensional if and only if X is finite. ) 4. (i) Consider then x n complex matrix (aij)· Prove that the mapping ~