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c0problems - FUNCTIONAL ANALYSIS LECTURE NOTES PROBLEMS ON...

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FUNCTIONAL ANALYSIS LECTURE NOTES: PROBLEMS ON c AND c 0 CHRISTOPHER HEIL Definition 1. Sequences with unspecified limits are indexed by the natural numbers N . We set c = c ( N ) = braceleftbig a = ( a k ) : lim k →∞ a k exists bracerightbig , c 0 = c 0 ( N ) = braceleftbig a = ( a k ) : lim k →∞ a k = 0 bracerightbig , c 00 = c 00 ( N ) = braceleftbig a = ( a k ) : only finitely many a k are nonzero bracerightbig . Definition 2 (Basis) . A countable sequence { x n } in a Banach space X is a basis for X if x X, unique scalars a n ( x ) such that x = summationdisplay n a n ( x ) x n . (1) We call the series in equation (1) the basis expansion or basis representation of x with respect to { x n } . Definition 3. Let { x n } be a basis for a Banach space X. (a) { x n } is an unconditional basis if the series in equation (1) converge unconditionally for each x X, i.e., if σ : N N is any bijection then the series a σ ( n )( x ) x σ ( n ) converges. A basis that is not an unconditional basis is called a conditional basis .
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