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Proof of the collections of sequences are linear spaces or vector space. 
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#1
Dec2408, 05:19 AM

P: 4

[b]1. The problem statement, all variables and given/known data[/b
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 XY=(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 


#2
Dec2408, 05:22 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,691

Okay, what have YOU done on this? I see no attempt at a solution. Also, you talk about "properties of theorem (**)" and "properties of (***) without saying what they are!



#3
Dec2408, 05:33 AM

P: 4

my project



#4
Dec2408, 07:35 AM

P: 4

Proof of the collections of sequences are linear spaces or vector space.
in theorem(**) there are properties about vector space
A1) x+y=y+x A2)(x+y)+z=x+(y+z) A3)0+x=x and x+0=x A4) u=(1)x satisfies x+u=0 M1)1x=x M2)b(cx)=(bc)x D)c(x+y)=cx+cy and (b+c)x=bx+cx 


#5
Dec2408, 07:48 AM

P: 4

theorem(***) is the norm properties
(i)lxl>=0; (ii)lxl=0 iff x=0 (iii)lcxl=lcl lxl (iv)l lxllyl l<=lx+yl<=lxl+lyl my problem is i dont`n know how i can explain these properties for a sequence, for example m is collection of all bounded sequences, c is collection of all convergent sequences how i show difference between them? thanks 


#6
Dec2408, 08:35 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,691




#7
Dec2408, 08:45 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,691




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