Give two examples of a vector space with infinite dimension

Give two examples of a vector space with infinite dimension. Give a basis for your examples.

Solution

A.

Vector space of all functions which can be represented by a power series.

Basis: {1,x,x^2,x^3,.....................,x^n,..................}

B.

Consider the set of sequences which are eventually zero. This is an infinite dimensional vector space with basis: {e_i:i=1,2,3,4,5.......................}

e_i=(0,,,,,,,,,,,,,,,,,,1,0,...)

ie 1 at ith place and o everywhere else.