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

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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 }.

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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 provided by:
Thokchom Sarojkumar Sinha, MSc
Education
  • BSc, Manipur University
  • MSc, Kanpur University
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