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Lets s denote the collection of all sequences in lR, let m denote the

collection of all bounded sequences in lR, let c denote the collection

of all convergent sequences in lR, and let Co denote the collection of

all sequences in lR which converge to zero.

(a)With the definition of sum given in (*) and the definition of product of

a sequence and real number given by a(xsubn)=(axsubn), show that each of

these collections has the properties of theorem(**). In each case the zero

element is the sequence teta=(0,0,...,0,..). (We sometimes say that these

collections are linear spaces or vector spaces.)

(b) If X=(xsubn) belongs to one of the collections m,c,csub0, define the norm

of X by lXl=sup{lxsubnl:nEN}. Show that thisnorm function has the properties

of (***). (For this reason, we sometimes say that these collections are

normed linear spaces.)

(*)definition:If X=(xsubn) and Y=(ysubn) are sequences in R to p, then we define their

sum to be the sequence X+Y=(xsubn + ysubn) in R to p, their difference

to be thesequence X-Y=(xsubn - xsubn), and their inner product to be the

sequence X.Y=(xsubn.ysubn) in R which is obtained by taking the inner

product of corresponding terms.

Similarly, if X=(xsubn) is a sequence in R

and if Y=(ysubn) is a sequence in R to p, we define the product of X and Y

to be the sequence in R to p denoted by XY=(xsubnysubn).

Finally,if Y=(ysubn) is a sequence in R with ysubn is not equal to 0, we

can define the quotient of a sequence X=(xsubn) in R to p by Y to be the

sequence X/Y=(xsubn/ysubn).

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Proof of the collections of sequences are linear spaces or vector space.

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