Kunze Linear Algebra · Problem 2.2.8

Vector Spaces

Mathematics — Methods · even-odd-function-subspaces

Problem

Let

V

be the vector space of all functions from

R

into

R

; let

V_e

be the subset of even functions,

f(-x) = f(x)

; let

V_o

be the subset of odd functions,

f(-x) = -f(x)

. (a) Prove that

V_e

and

V_o

are subspaces of

V

. (b) Prove that

V_e + V_o = V

. (c) Prove that

V_e \cap V_o = \{0\}