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Chapter 2: Problem 5

Show that \(C^{\infty}\) is a subspace of \(\mathcal{F}(R, C)\).

Short Answer

Expert verified
The set \(C^{\infty}\) is a subspace of \(\mathcal{F}(\mathbb{R}, \mathbb{C})\) because it contains the zero vector (the zero function), is closed under addition (the sum of any two infinitely differentiable functions is also infinitely differentiable), and is closed under scalar multiplication (the product of an infinitely differentiable function and a scalar is also infinitely differentiable).

Step by step solution

01

Check if the zero vector is in \(C^{\infty}\)

The zero vector in \(\mathcal{F}(\mathbb{R}, \mathbb{C})\) is the zero function, which maps every real number to the complex number 0. The zero function is clearly infinitely differentiable (since all its derivatives are also the zero function), so the zero vector is in \(C^{\infty}\).
02

Check if the sum of functions in \(C^{\infty}\) is also in \(C^{\infty}\)

Let \(f\) and \(g\) be two functions in \(C^{\infty}\). We need to show that their sum, denoted by \(h(x) = f(x) + g(x)\), is also in \(C^{\infty}\). Since both \(f\) and \(g\) are infinitely differentiable, their sum also has infinitely many derivatives: \((f+g)^{(n)}(x) = f^{(n)}(x) + g^{(n)}(x)\) for all \(x \in \mathbb{R}\) and for all \(n \in \mathbb{N}\) (including \(n = 0\), which represents the original functions). Therefore, \(h(x) = f(x) + g(x)\) is in \(C^{\infty}\).
03

Check if the product of a function in \(C^{\infty}\) and a scalar is also in \(C^{\infty}\)

Let \(f\) be a function in \(C^{\infty}\) and let \(c \in \mathbb{C}\) be a scalar. We need to show that the product \(cf\) is also in \(C^{\infty}\). Since \(f\) is infinitely differentiable, the product function \(cf(x) = c \cdot f(x)\) also has infinitely many derivatives: \((cf)^{(n)}(x) = c \cdot f^{(n)}(x)\) for all \(x \in \mathbb{R}\) and for all \(n \in \mathbb{N}\). Therefore, \(cf\) is in \(C^{\infty}\). Since all three properties of vector subspaces are satisfied, we conclude that \(C^{\infty}\) is a subspace of \(\mathcal{F}(\mathbb{R}, \mathbb{C})\).

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Most popular questions from this chapter

Refer to the following matrices: $$A=\left[\begin{array}{rr} 1 & 2 \\ 3 & -4 \end{array}\right], \quad B=\left[\begin{array}{rr} 5 & 0 \\ -6 & 7 \end{array}\right], \quad C=\left[\begin{array}{rrr} 1 & -3 & 4 \\ 2 & 6 & -5 \end{array}\right], \quad D=\left[\begin{array}{rrr} 3 & 7 & -1 \\ 4 & -8 & 9 \end{array}\right]$$ Find \(\quad(a) A^{T}\) (b) \(B^{T}\) \((\mathrm{c})(A B)^{T}\) (d) \(A^{T} B^{T}\). [Note that \(A^{T} B^{T} \neq(A B)^{T}\).]

Prove that if \(\\{x, y\\}\) is a basis for a vector space over \(C\), then so is $$ \left\\{\frac{1}{2}(x+y), \frac{1}{2 i}(x-y)\right\\} . $$

Refer to the following matrices: $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 3 & 4 \end{array}\right], \quad B=\left[\begin{array}{rrr} 4 & 0 & -3 \\ -1 & -2 & 3 \end{array}\right], \quad C=\left[\begin{array}{rrrr} 2 & -3 & 0 & 1 \\ 5 & -1 & -4 & 2 \\ -1 & 0 & 0 & 3 \end{array}\right], \quad D=\left[\begin{array}{rr} 2 \\ -1 \\ 3 \end{array}\right].$$ Find (a) \(3 A-4 B,\) (b) \(A C,\) (c) \(B C,\) (d) \(A D,\) (e) \(B D,\) (f) \(C D.\)

Refer to the following matrices: $$A=\left[\begin{array}{rr} 1 & 2 \\ 3 & -4 \end{array}\right], \quad B=\left[\begin{array}{rr} 5 & 0 \\ -6 & 7 \end{array}\right], \quad C=\left[\begin{array}{rrr} 1 & -3 & 4 \\ 2 & 6 & -5 \end{array}\right], \quad D=\left[\begin{array}{rrr} 3 & 7 & -1 \\ 4 & -8 & 9 \end{array}\right]$$ Find (a) \(A^{2}\) and \(A^{3}\) (c) \(C D\) (b) \(A D\) and \(B D\).

Let \(V\) and \(W\) be nonzero vector spaces over the same field, and let \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) be a linear transformation. (a) Prove that \(T\) is onto if and only if \(T^{t}\) is one-to-one. (b) Prove that \(T^{t}\) is onto if and only if \(T\) is one-to-one.
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