Proof of the collections of sequences are linear spaces or vector space.

In summary: They are NOT different. The entire point is that the sets of all "bounded sequences", "convergent sequences", and "sequences converging to 0" are all the same set- that is, there is no difference between them. That is what you are to show.
  • #1
irmctn
4
0
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).


Homework Equations





The Attempt at a Solution

 
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  • #2
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
my project
 

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  • #4
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
theorem(***) is the norm properties
(i)lxl>=0;
(ii)lxl=0 iff x=0
(iii)lcxl=lcl lxl
(iv)l lxl-lyl l<=lx+-yl<=lxl+lylmy 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
irmctn said:
in theorem(**) there are properties about vector space
A1) x+y=y+x
Here, x is a sequence {xn} and y is a sequence {xy}. According to your definitions, x+ y is the sequence {xn+ yn} and y+ x is the sequence {yn+ xn}. Are those two sequences equal?

A2)(x+y)+z=x+(y+z)
With x and y as above, z is the sequence {zn}. x+ y is the sequence {xn+ yn} as before so (x+y)+ z is the sequence {(xn+ Yn)+ zn}. For the same reasons, x+ (y+ z) is the sequence {xn+ (yn+ zn)}. Are those two sequences equal?

A3)0+x=x and x+0=x
There exist 0 such that... Suppose 0 is the sequence {0} (for all n). What is the sequence 0+ x? x+ 0?

A4) u=(-1)x satisfies x+u=0
If x is the sequence {xn}, what is (-1)x?

M1)1x=x
If x is the sequence {xn}, what is 1x?

M2)b(cx)=(bc)x
If x is the sequence {xn}, cx is the sequence {cxn} so b(cx) is the sequence {b(cxn)} and (bc)x is the sequence {(bc)xn}. Are those two sequences the same?

D)c(x+y)=cx+cy and (b+c)x=bx+cx
If x is the sequence {xn} and y is the sequence {yn}. What is c(x+y)? What is cx+ cy? Are they the same?
 
  • #7
irmctn said:
theorem(***) is the norm properties
You are told that if x is the sequence {xn}, |x| is the supremum (least upper bound) of the absolute value all numbers in that sequence. Since these are bounded sequences, that supremum exists.

(i)lxl>=0;
If x is the sequence {xn}, |xn| is never negative so its suprememum can't be negative.

(ii)lxl=0 iff x=0
The "if" part should be trivial. If |x|= 0, then 0 is an upper bound for the absolute values of the numbers in the sequence: |xn<= 0 for all n. But an absolute value cannot be negative

(iii)lcxl=lcl lxl
If x is the sequence {xn}, what is cx? What is |cx| then?

(iv)l lxl-lyl l<=lx+-yl<=lxl+lyl
Do you know that |x|- |y|<= |x+ y| for NUMBERS? And so for each term of the sequences {xn} and {yn}?


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
You don't need to show any difference between them.
 

1. What is a linear space or vector space?

A linear space or vector space is a mathematical structure that consists of a set of objects called vectors, along with operations such as addition and scalar multiplication that can be performed on these vectors. It follows certain axioms or rules, such as closure under addition and scalar multiplication, to define a linear combination and ensure that the space is closed and consistent.

2. How do you prove that a collection of sequences is a linear space or vector space?

To prove that a collection of sequences is a linear space or vector space, we need to show that it satisfies all the axioms or rules of a linear space. This includes closure under addition and scalar multiplication, existence of a zero vector and additive inverses, and scalar associativity and distributivity. By demonstrating that these axioms hold for the collection of sequences, we can prove that it is a linear space or vector space.

3. Why is it important to prove that a collection of sequences is a linear space or vector space?

Proving that a collection of sequences is a linear space or vector space is important because it provides a rigorous mathematical foundation for analyzing and manipulating these sequences. It allows us to apply techniques and principles from linear algebra to solve problems and make predictions based on these sequences.

4. Can a collection of sequences be a linear space or vector space if it does not satisfy all the axioms?

No, in order for a collection of sequences to be considered a linear space or vector space, it must satisfy all the axioms or rules. If even one of the axioms is not satisfied, the collection of sequences cannot be considered a linear space or vector space.

5. Are there any real-world applications of proving that a collection of sequences is a linear space or vector space?

Yes, there are many real-world applications of proving that a collection of sequences is a linear space or vector space. This includes applications in fields such as physics, engineering, and computer science, where sequences are used to model and analyze various phenomena. By proving that a collection of sequences is a linear space or vector space, we can apply powerful mathematical tools and techniques to understand and manipulate these sequences in a meaningful way.

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