Let R infinity = {(x 1, x 2, x 3, ) : x k R for all k} be the vector space of al
ID: 3099507 • Letter: L
Question
Let R infinity = {(x 1, x 2, x 3, ) : x k R for all k} be the vector space of all real sequences. Determine whether the following subsets are subspaces of R infinity. No justification is required, just a yes or no answer. The sequences are described using words, because reading comprehension is part of the problem. All sequence whose terms are zero infinitely often. All sequences whose terms are zero from some point onward. All sequences whose terms arc non-increasing. All geometric sequences. All sequences that have a limit. All sequences whose sum forms a convergent series.Explanation / Answer
a. No. Counter example: (0,1,0,1,0,1,...) and (1,0,1,0,1,0,...) have infinitely many 0s, but the sum of them is (1,1,1,1,1,1,...) which has no zero and hence is not in the set.
b. Yes. Let a be a sequence with all 0s from m_th position onwards, let b be a sequence with all 0s from n_th position onwards. Without loss of generality, assume m > n, then any linear combinations of a and b still have all 0s from m_th position onwards.
c. Yes. Let m=(a1,a2, ...), n=(b1,b2, ...) such that a1 >= a2 >= ... and b1 >= b2 >= .., then xa1 + ya2 >= xa2 + yb2 >= ... for any real number x,y.Hence, xm + yn is in the set.
d. Yes. Let x be a geometric sequence with common difference d1, y be a geometric sequence with common difference d2. Then ax + by wil be a geometric sequence with common difference d1+d2, for any nonzero real number a and b.
e. Yes. If x1 converges to a, x2 converges to b, then the linear combination of x1 and x2 will converge to the same linear combination of a and b.
f. Yes. Similar proof as (e).
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