--- Commentary: $\textbf{Exercise 19.}$ Prove that the real vector space of all continuous real-valued functions on the interval $[0, 1]$ is infinite-dimensional. $\textbf{Solution 19.}$ Let $V$ denote the real vector space of all continuous real-valued functions on the interval $[0, 1]$. For each positive integer $m$, the list $1, x, \ldots, x^m$ is linearly independent in $V$ (because if $a_0, \ldots, a_m \in \mathbb{R}$ are such that $a_0 + a_1x + \cdots + a_mx^m = 0$ for every $x \in [0, 1]$, then the polynomial above has infinitely many zeros and hence all its coefficients equal 0). The existence of a spanning list in $V$ would contradict 2.22. Thus $V$ is infinite-dimensional. $\textit{Commentary:}$ This exercise asks to prove that a specific function space, namely the space of continuous real-valued functions on $[0, 1]$, is infinite-dimensional. The proof again uses the characterization of infinite-dimensional spaces from Exercise 17. It exhibits an infinite sequence of functions in $V$ with the property that any initial segment is linearly independent. The sequence used is the sequence of monomials $1, x, x^2, \ldots$. To see that any finite subset of these functions is linearly independent, suppose a linear combination of them equals the zero function. This linear combination is a polynomial function. But a nonzero polynomial can have only finitely many zeros, while the zero function on $[0, 1]$ has infinitely many zeros. Thus, the polynomial must be the zero polynomial, which means all its coefficients must be zero. This shows that the monomials are linearly independent. The proof then argues by contradiction that $V$ cannot have a finite spanning set, using the result from 2.22. This exercise reinforces the concept of linear independence for functions and provides practice in reasoning about function spaces and polynomials. $\textit{Additional Examples:}$ The space of continuous functions $C([a, b], \mathbb{R})$ on any closed interval $[a, b]$ is infinite-dimensional. The sequence of monomials $1, x, x^2, \ldots$ is linearly independent. The space of differentiable functions $C^1([0, 1], \mathbb{R})$ is infinite-dimensional. The sequence of monomials $1, x, x^2, \ldots$ is linearly independent. The space of analytic functions on $[0, 1]$ (i.e., functions that can be represented by a power series) is infinite-dimensional. The sequence of monomials $1, x, x^2, \ldots$ is linearly independent. The space of continuous functions $C([0, 1], \mathbb{C})$ with complex values is infinite-dimensional over $\mathbb{C}$. The sequence of monomials $1, z, z^2, \ldots$ is linearly independent. These examples illustrate that many common function spaces are infinite-dimensional. This is true for continuous functions, differentiable functions, analytic functions, and complex-valued functions. In each case, the monomials form an infinite linearly independent set, showing that the space has infinite dimension. This is a fundamental property of function spaces that distinguishes them from finite-dimensional vector spaces. ---