Final answer:
C¹ [a, b], the set of functions with continuous first derivatives on an interval [a, b], satisfies all three criteria for being a subspace of the vector space C[a, b], which includes containing the zero vector and being closed under vector addition and scalar multiplication.
Step-by-step explanation:
The question asks whether the set C¹ [a, b] of functions with continuous first derivatives on the interval [a, b] is a subspace of the vector space C[a, b], where C[a, b] is the space of continuous functions on [a, b]. To determine if C¹ [a, b] is a subspace, we must check if it satisfies three criteria: it must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication.
Firstly, the zero vector in this context is the zero function, which has a continuous first derivative (which is also the zero function) and thus belongs to C¹ [a, b]. Secondly, if y(x) and z(x) are two functions in C¹ [a, b], their sum y(x) + z(x) will also be in C¹ [a, b], since the sum of two continuous functions is continuous and the derivative of the sum is the sum of the derivatives, which are both continuous. Lastly, for any scalar λ, multiplying a function y(x) by λ yields λ y(x), which will still have a continuous derivative λ dy(x)/dx. Thus, the set C¹ [a, b] is closed under scalar multiplication.
Given these properties, we can conclude that the set C¹ [a, b] is indeed a subspace of C[a, b].