Are the odd real functions a subspace Prove or give countere

Are the odd real functions a subspace? Prove or give counter-example.

Solution

The odd real functions will form a subspace if

a) The zero function must be present in the subspace

b) For function x and y belonging to subspace, (x+y) belongs to subspace

c) For x belonging to subspace, cx will also belong to subspace where c is a constant

Proof of part a

Since 0(x) = 0(-x), where 0 is a zero function

Proof of part b

f(x) = -f(-x)

g(x) = -g(-x)

f(x) + g(x) = -f(-x) - g(-x) = -(f(-x) + g(-x))

Hence the function is an odd function, which will still be present in the subspace

Proof of part C

f(x) = -f(-x)

cf(x) = -cf(-x)

Hence it is also an odd function, therefore belongs to subspace