# 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. Prove that V is a vector space over F.

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.

Prove that V is a vector space over F.

See attached file for full problem description.

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#### Solution Summary

This solution is comprised of a detailed explanation of a vector space over the field real numbers. It contains step-by-step explanation for the following problem:

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 belings to F, where equality, addition and scalar multiplication are defined componentwise.

Prove that V is a vector space over F.

Notes are also given at the end.

Solution contains detailed step-by-step explanation.