91Ó°ÊÓ

#tag_title#Prove even and odd functions are subspaces# #tag_content#To prove that the set of even functions and the set of odd functions are subspaces of \(\mathcal{F}\left(F_{1}, F_{2}\right)\), we will show each of the three required properties. 1. Zero element: The zero function \(z(t) = 0\) is both an even function and an odd function, satisfying the first property for both sets. This is because \(z(-t) = 0 = z(t)\) and \(z(-t) = 0 = -z(t)\). 2. Closure under addition: Let \(f\) and \(g\) be even functions. Then, for any \(t \in F_{1}\), \((f+g)(-t) = f(-t) + g(-t) = f(t) + g(t) = (f+g)(t)\), so \(f+g\) is an even function. Similarly, if \(f\) and \(g\) are odd functions, \((f+g)(-t) = f(-t) + g(-t) = -f(t) - g(t) = -(f(t) + g(t)) = -(f+g)(t)\), so \(f+g\) is an odd function. 3. Closure under scalar multiplication: Let \(f\) be an even function and \(c \in F_{2}\). Then, for any \(t \in F_{1}\), \((cf)(-t) = c \cdot f(-t) = c \cdot f(t) = (cf)(t)\), so \(cf\) is an even function. Similarly, if \(f\) is an odd function, \((cf)(-t) = c \cdot f(-t) = c \cdot (-f(t)) = - (c \cdot f(t)) = -(cf)(t)\), so \(cf\) is an odd function. Therefore, both the set of even functions and the set of odd functions are subspaces of \(\mathcal{F}\left(F_{1}, F_{2}\right)\), since they satisfy the three required properties.