91Ó°ÊÓ

The zero function is the function \(f(x) = 0\) for all x in the interval [a, b]. Since the zero function is continuous and has derivatives of all orders equal to zero, it is n-times continuously differentiable and belongs to \(C^{n}[a, b]\).

Let \(f(x)\) and \(g(x)\) be two functions in \(C^{n}[a, b]\). This means that both functions are n-times continuously differentiable on the interval [a, b]. We need to show that their sum, \(h(x) = f(x) + g(x)\), is also n-times continuously differentiable. Since both functions are n-times continuously differentiable, we have: \(\frac{d^k}{dx^k}f(x)\) and \(\frac{d^k}{dx^k}g(x)\) exist and are continuous for all \(k\) in the range \(\{0,1,2,...,n\}\). Now, consider their sum: \(h(x) = f(x) + g(x)\) Taking the k-th derivative of h(x): \(\frac{d^k}{dx^k}h(x) = \frac{d^k}{dx^k}(f(x) + g(x))\) Using the linearity of the differentiation operator: \(\frac{d^k}{dx^k}h(x) = \frac{d^k}{dx^k}f(x) + \frac{d^k}{dx^k}g(x)\) Since sum of two continuous functions is continuous, \(\frac{d^k}{dx^k}h(x)\) is continuous for all k in the range \(\{0,1,2,...,n\}\), so h(x) is n-times continuously differentiable. Therefore, \(C^{n}[a, b]\) is closed under addition.

Let \(f(x)\) be in \(C^{n}[a, b]\) and let c be a scalar. We need to show that \(cf(x)\) is also n-times continuously differentiable. Since \(f(x)\) is n-times continuously differentiable, its k-th derivative \(\frac{d^k}{dx^k}f(x)\) exists and is continuous for all \(k\) in the range \(\{0,1,2,...,n\}\). Now, consider the scalar multiple \(cf(x)\). Taking the k-th derivative of \(cf(x)\), we have: \(\frac{d^k}{dx^k}(cf(x)) = c\frac{d^k}{dx^k}f(x)\) Since multiplication by a scalar preserves continuity, the k-th derivative of \(cf(x)\) is continuous for all \(k\) in the range \(\{0,1,2,...,n\}\). Thus, \(cf(x)\) is n-times continuously differentiable, and so \(C^{n}[a, b]\) is closed under scalar multiplication. In conclusion, \(C^{n}[a, b]\) is a subspace of \(C[a, b]\) because it contains the zero function and is closed under both addition and scalar multiplication.