Let R denote the vector space of all sequences of real numbers. (Addition and scalar multiplication are defined coordinatewise.) In each of the following a subset of Ris described. Write yes if the set is a subspace of R and no if it is not. (a) Sequences that have infinitely many zeros (for example, (1, 1, 0, 1, 1, 0, 1, 1, 0, . . . )). Answer: (b) Sequences which are eventually zero. (A sequence (T) is eventually zero if there is an index no such that In = 0 whenever n ≥ no.) Answer: (c) Sequences that are absolutely summable. (A sequence (Tk) is absolutely summable if Σ1k|<∞o.) Answer: (d) Bounded sequences. (A sequence (Tk) is bounded if there is a positive number M such that TM for every k.) Answer: (e) Decreasing sequences. (A sequence (T) is decreasing if In+1 ≤ In for each n.) Answer: (f) Convergent sequences. Answer: (g) Arithmetic progressions. (A sequence (Tk) is arithmetic if it is of the form (a, a + k, a + 2k, a + 3k,...) for some constant k.) Answer: (h) Geometric progressions. (A sequence (Tk) is geometric if it is of the form (a, ka, k2a, k³ a, ...) for some constant k.) Answer:

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