let V be the set of all real-valued continuous functions defined on R1. If f and g are in V, we define f ¨’ g by (f ¨’ g)(t) = f(t) + g(t). If f is in V and c is a scalar, we define c f by (c f)(t) = cf(t). Then V is a vector space, which is denoted by C(−¡Þ, ¡Þ).
Which of the following subsets are subspaces of the vector space C(−¡Þ, ¡Þ)? (Note: Calculus is required to complete this exercise.)
(a) All nonnegative functions
(b) All constant functions
To determine whether a subset is a subspace of a vector space, we need to check whether it satisfies three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector.
(a) All nonnegative functions:
To check closure under addition, consider two nonnegative functions, f and g. For any real number t, (f + g)(t) = f(t) + g(t). Since both f(t) and g(t) are nonnegative, their sum will also be nonnegative. Therefore, the subset of all nonnegative functions is closed under addition.
To check closure under scalar multiplication, consider a nonnegative function f and a scalar c. For any real number t, (cf)(t) = c * f(t). Since f(t) is nonnegative, and multiplying a nonnegative number by any scalar will also give a nonnegative result, the subset is closed under scalar multiplication.
To check if the subset contains the zero vector, we need to find a function that is nonnegative for all values of t. The zero function, f(t) = 0 for all t, satisfies this condition. Therefore, the subset of all nonnegative functions is a subspace of the vector space C(−∞, ∞).
(b) All constant functions:
To check closure under addition, consider two constant functions, f(t) = a and g(t) = b (where a and b are constants). (f + g)(t) = a + b, which is also a constant function. Therefore, the subset of all constant functions is closed under addition.
To check closure under scalar multiplication, consider a constant function f(t) = a and a scalar c. (cf)(t) = c * a, which is also a constant function. Therefore, the subset of all constant functions is closed under scalar multiplication.
To check if the subset contains the zero vector, we need to find a constant function that is in the subset. The constant function f(t) = 0 for all t satisfies this condition. Therefore, the subset of all constant functions is a subspace of the vector space C(−∞, ∞).
In summary, both (a) the subset of all nonnegative functions and (b) the subset of all constant functions are subspaces of the vector space C(−∞, ∞).