2. Let V be the vector space over C consisting of all infinite sequences (01, 02, 03, ... ) with only finitely many nonz

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2 Let V Be The Vector Space Over C Consisting Of All Infinite Sequences 01 02 03 With Only Finitely Many Nonz 1 (95.87 KiB) Viewed 69 times

2. Let V be the vector space over C consisting of all infinite sequences (01, 02, 03, ... ) with only finitely many nonzero entries. Let eį be the sequence with only one nonzero entry being 1 at i-th position. It follows that B := {C1, C2, ... } is a basis for V. Show that, however, ß* = {et, e, ... } is NOT a basis for V*. (Hint. Let f EV* defined by f(a1, 42, ...) Lieldi. Explain that f is well-defined and show that f & span 8*. Note that in a vector space, “span” only collects finite linear combinations.) Remark. If dim V = 0, one can show that dim V* > dim V.