--- Commentary: $\textbf{Exercise 24.}$ A function $f: \mathbb{R} \to \mathbb{R}$ is called even if $f(-x) = f(x)$ for all $x \in \mathbb{R}$. A function $f: \mathbb{R} \to \mathbb{R}$ is called odd if $f(-x) = -f(x)$ for all $x \in \mathbb{R}$. Let $V_e$ denote the set of real-valued even functions on $\mathbb{R}$ and let $V_o$ denote the set of real-valued odd functions on $\mathbb{R}$. Show that $\mathbb{R}^\mathbb{R} = V_e \oplus V_o.$ $\textbf{Solution 24.}$ Suppose $f \in V_e \cap V_o$. Then for all $x \in \mathbb{R}$ we have $f(x) = f(-x) = -f(x),$ where the first equality holds because $f$ is even and the second equality holds because $f$ is odd. The equation above implies that $f$ is the 0 function. Thus by 1.46, $V_e + V_o$ is a direct sum. Suppose $g \in \mathbb{R}^\mathbb{R}$. Then $g(x) = \frac{g(x) + g(-x)}{2} + \frac{g(x) - g(-x)}{2}$ for every $x \in \mathbb{R}$. With functions $g_e$ and $g_o$ defined as in the equation above, we have $g = g_e + g_o$, where $g_e \in V_e$ and $g_o \in V_o$. Thus $\mathbb{R}^\mathbb{R} = V_e \oplus V_o$. $\textit{Commentary:}$ This exercise demonstrates a fundamental decomposition of the space of real-valued functions on $\mathbb{R}$ into the direct sum of the subspaces of even and odd functions. The proof first shows that the only function that is both even and odd is the zero function, which means that $V_e$ and $V_o$ intersect only at 0. This establishes that $V_e + V_o$ is a direct sum. Next, it shows that any function $g$ can be uniquely written as the sum of an even function $g_e$ and an odd function $g_o$. The even part $g_e$ is constructed by taking the average of $g(x)$ and $g(-x)$, while the odd part $g_o$ is constructed by taking half of the difference of $g(x)$ and $g(-x)$. This establishes that $\mathbb{R}^\mathbb{R} = V_e \oplus V_o$. This decomposition is a powerful tool in functional analysis and has many applications, particularly in the study of Fourier series and Fourier transforms. It allows any function to be split into a part that is symmetric about the origin and a part that is antisymmetric. $\textit{Additional Examples:}$ The space of continuous functions $C(\mathbb{R}, \mathbb{R})$ can be decomposed as $C(\mathbb{R}, \mathbb{R}) = C_e(\mathbb{R}, \mathbb{R}) \oplus C_o(\mathbb{R}, \mathbb{R})$, where $C_e(\mathbb{R}, \mathbb{R})$ and $C_o(\mathbb{R}, \mathbb{R})$ are the subspaces of even and odd continuous functions respectively. The space of square-integrable functions $L^2(\mathbb{R}, \mathbb{C})$ can be decomposed as $L^2(\mathbb{R}, \mathbb{C}) = L^2_e(\mathbb{R}, \mathbb{C}) \oplus L^2_o(\mathbb{R}, \mathbb{C})$, where $L^2_e(\mathbb{R}, \mathbb{C})$ and $L^2_o(\mathbb{R}, \mathbb{C})$ are the subspaces of even and odd square-integrable functions respectively. The space of polynomials $\mathbb{R}[x]$ can be decomposed as $\mathbb{R}[x] = \mathbb{R}[x]_e \oplus \mathbb{R}[x]_o$, where $\mathbb{R}[x]_e$ and $\mathbb{R}[x]_o$ are the subspaces of even and odd polynomials respectively. The space of trigonometric polynomials (linear combinations of sines and cosines) can be decomposed into the direct sum of the subspace of cosine polynomials (even) and the subspace of sine polynomials (odd). These examples illustrate that the decomposition into even and odd parts is a general phenomenon that applies to many function spaces. The specific details of the spaces may vary (e.g., continuous functions, square-integrable functions, polynomials), but the underlying principle remains the same. This demonstrates the power and universality of the concept of direct sum decompositions in the study of function spaces.