Vector Space of Bounded Sequences

In summary, we are considering the vector space l_infty(R) of all bounded sequences and determining whether the norms || ||_# = |x_1| and || ||_infty = sup |x_n| for n in N are defined on this space. We verify that || ||_# is defined by showing that it satisfies the axioms of a norm, but for || ||_infty we are unsure how to use the triangle inequality.
  • #1
bugatti79
794
1

Homework Statement



Consider the vector space l_infty(R) of all bounded sequences. Decide whether or not the following norms are defined on l_infty(R) . If they are, verify by axioms. If not, provide counter example.

Homework Equations



x in l_infty(R); x=(x_n), (i) || ||_# defined by ||x||_#=|x_1|, (ii) |\ ||_infty defined by || ||_infty =sup |x_n| for n in N

The Attempt at a Solution



(i)
a) Since x is in l_infty(R), there exist k_x >0 s.t |x_n| <=k_x for all n in N
y is in l_infty(R), there exist k_y>0 s.t |y_n| <=k_y for all n in N

||x+y||_#=||(x_n+y_n)||_#=|x_n+y_n|. Now we have that |a+b| <=|a|+|b|, therefore
<=|x_n|+|y_n|
<=k_x+k_y for all n in N
<=||x||_#+||y||_#

b) ||ax||_#=|ax_1|
=|a| |x_1|
=|aA ||x||_#

Axioms c) and d) I don't know how to attempt for this space

(ii) || ||_infty =sup |x_n| for n in N

Let x be in l_infty(R), x=(x_1)

a) ||x||_infty>=sup|x_n|>=0 where sup|x_n|>-0 for n in N for all x_n in l_infty(R)
b) ||x||=0 iff x=0
c) ||ax||_infty =sup|ax_n|
=|a| sup|x_n| for n in N
=|a| ||x||_infty

d) Now sure how to use the trinagle inequality for this space..?
 
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  • #2
Any review of this folks?
 

FAQ: Vector Space of Bounded Sequences

1. What is a vector space of bounded sequences?

A vector space of bounded sequences is a collection of all bounded sequences of real numbers with operations of addition and scalar multiplication defined on them. It is a mathematical concept used in functional analysis and linear algebra.

2. How is a bounded sequence defined?

A bounded sequence is a sequence of real numbers that is limited within a certain range, meaning that the values of the sequence do not grow infinitely large. It is often denoted by the symbol (xn).

3. What are the properties of a vector space of bounded sequences?

Some of the properties of a vector space of bounded sequences include closure under addition and scalar multiplication, existence of a zero vector and additive inverse, and the associative, commutative, and distributive properties.

4. How is a vector space of bounded sequences useful in mathematics?

A vector space of bounded sequences is useful in mathematics for studying and analyzing functions and their properties. It is also used in solving systems of linear equations and in the study of infinite-dimensional spaces.

5. Can a vector space of bounded sequences have infinite dimensions?

Yes, a vector space of bounded sequences can have infinite dimensions. This is because the set of all bounded sequences is infinite and the number of dimensions depends on the number of elements in the basis of the vector space.

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