Question

Consider the vector space V of all function from R to R. Define T:V→V by setting T(f)(t)=21​f(t)+21​f(−t) for every f∈V and every t∈R. (a) Show that T is a projection. (b) Show that Range (T) is the set of all even functions in V, and that Null(T) is the set of all odd functions in V.

Answer 1

To show that T is a projection, we need to prove two conditions: T is idempotent and T is linear. To show Range(T) is the set of all even functions and Null(T) is the set of all odd functions, we need to show that every even function f ∈ Range(T) satisfies T(f)(t) = f(t) for every t ∈ ℝ and every odd function f ∈ Null(T) satisfies T(f)(t) = 0 for every t ∈ ℝ.

Step-by-step explanation:

To show that T is a projection, we need to prove two conditions:

1. T is idempotent: T(T(f))(t) = T(f)(t) for every f ∈ V and every t ∈ ℝ.
2. T is linear: T(f+g)(t) = T(f)(t) + T(g)(t) and T(cf)(t) = cT(f)(t) for every f, g ∈ V, c ∈ ℝ, and every t ∈ ℝ.

To show Range(T) is the set of all even functions and Null(T) is the set of all odd functions, we need to show that:

1. Every even function f ∈ Range(T) satisfies T(f)(t) = f(t) for every t ∈ ℝ.
2. Every odd function f ∈ Null(T) satisfies T(f)(t) = 0 for every t ∈ ℝ.