Explore BrainMass

# Prove that U is a Subspace of V and is Contained in W

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Please view the attached file for the full solution. What is presented below has many missing parts as the full question could not be copied properly.

Let F be the field of real numbers and let V be the set of all sequences:

( a_1, a_2, ... , a_n, ... ), a_i belongs to F, where equality, addition and scalar multiplication are defined component wise. Then V is a vector space over F.

Let W = {( a_1, a_2, ... , a_n, ... ) belongs to V such that lim a_n = 0 where n tends to infinity }. Then W is a subspace of V.

Let U = {( a_1, a_2, ... , a_n, ... ) belongs to V such that summation (a_i)^2 is finite}

Prove that U is a subspace of V and is contained in W,
where W = {( a_1, a_2, ... , a_n, ... ) belongs to V such that lim a_n = 0 where n tends to infinity }.

https://brainmass.com/math/vector-calculus/prove-subspace-contained-138360

#### Solution Preview

Thanks for using BrainMass.

Let be the field of real numbers and let be the set of all sequences , , where equality,
addition and scalar multiplication are defined component wise. Then is a vector space over .
Let . Then is a subspace of .
Let .
Prove that is a subspace of and is contained in , where .

Solution:- Let .
We have to prove that is a subspace of .
Let and let such that
where is finite and where is finite.
Then

where
For
...

#### Solution Summary

This solution is comprised of a detailed, step by step response which illustrates how to solve this complex math problem related to subsets. The complete solution is provided in an attached Word file.

\$2.49