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irmctn
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1. Homework Statement [/b
Let's 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).
Let's 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).