# Help with a subspace problem

## Main Question or Discussion Point

Let V be the vector space of all functions from R to R, equipped with the usual
operations of function addition and scalar multiplication. Let E be the subset of even
functions, so E = {f $$\epsilon$$ V |f(x) = f(−x), $$\forall$$x $$\epsilon$$ R} , and let O be the subset of odd
functions, so that O = {f $$\epsilon$$ V |f(x) = −f(−x), $$\forall$$x $$\epsilon$$ R} . Prove that:
(a) E and O are subspaces of V .
(b) E $$\cap$$ O = {0}.
(c) E + O = V .

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This is a good problem.

I don't think this should be too difficult for you. I think all you need is a few tips to help you get it. How about you show us what you have done so far?

Basically, all you need to do is apply your definitions. and use one of your proof techniques. For example, in "b" try using proof by contradiction. So assume that th intersection of E and O is not empty, and see where this leads you.

Give the problem a try or tell us your ideas at least and then someone will give you some good comments and feedback. :)

i did a)
trying b and see where it leads

thanks

for b)

i said
E$$\cap$$O= {f$$\epsilon$$v | f(x)= f(-x) and f(x)=-f(-x), $$\forall$$x$$\epsilon$$R}
E$$\cap$$O= {f$$\epsilon$$v | f(x)= -f(x), $$\forall$$x$$\epsilon$$R}
E$$\cap$$O= {f$$\epsilon$$v | f(x)= 0, $$\forall$$x$$\epsilon$$R}
E$$\cap$$O= {0} (f(x) =0 is the zero function)

what do you think?

and how about c) any idea??

HallsofIvy
Homework Helper
Be careful with c) E+ O is the space spanned by the vectors in E and in O. In other words, functions in it can be written as a linear combination of even and odd functions.

a) $f_E(x)= (f(x)+ f(-x))/2$, the even "part" of f
b) $f_O(x)= (f(x)- f(-x))/2$, the odd "part" of f

Can you see that $f_E$ is an even function and that $f_O$ is an odd function? And that $f(x)= f_E(x)+ f_O(x)$? What does that tell you about part c?

for b)

i said
E$$\cap$$O= {f$$\epsilon$$v | f(x)= f(-x) and f(x)=-f(-x), $$\forall$$x$$\epsilon$$R}
E$$\cap$$O= {f$$\epsilon$$v | f(x)= -f(x), $$\forall$$x$$\epsilon$$R}
E$$\cap$$O= {f$$\epsilon$$v | f(x)= 0, $$\forall$$x$$\epsilon$$R}
E$$\cap$$O= {0} (f(x) =0 is the zero function)

what do you think?