--- Commentary: $\textbf{Exercise 18.}$ Prove that $\mathbb{F}^\infty$ is infinite-dimensional. $\textbf{Solution 18.}$ For each positive integer $m$, let $e_m$ be the element of $\mathbb{F}^\infty$ whose $m$th coordinate equals 1 and whose other coordinates equal 0: $e_m = (0, \ldots, 0,1, 0, \ldots).$ $\uparrow$ $m$th coordinate Then $e_1, \ldots, e_m$ is a linearly independent list of vectors in $\mathbb{F}^\infty$, as is easy to verify. Exercise 17 now implies that $\mathbb{F}^\infty$ is infinite-dimensional. $\textit{Commentary:}$ This exercise asks to prove that a specific vector space, namely $\mathbb{F}^\infty$ (the space of infinite sequences with entries from $\mathbb{F}$), is infinite-dimensional. The proof uses the characterization of infinite-dimensional spaces from Exercise 17. It exhibits an infinite sequence of vectors in $\mathbb{F}^\infty$ with the property that any initial segment is linearly independent. The sequence used is the standard sequence of unit vectors $e_1, e_2, \ldots$, where $e_m$ has a 1 in the $m$th coordinate and 0s everywhere else. It's easy to see that any finite subset of these vectors is linearly independent, because a linear combination of them that equals zero must have all coefficients equal to zero (look at each coordinate separately). This proof is a standard way to show that sequence spaces like $\mathbb{F}^\infty$ are infinite-dimensional. The unit vectors form a natural linearly independent set that can be extended indefinitely. The exercise provides practice in applying the characterization of infinite-dimensional spaces from the previous exercise. It also reinforces the concept of the unit vectors and their linear independence. $\textit{Additional Examples:}$ The space $(\mathbb{Q}_p)^\infty$ (infinite sequences with entries from the $p$-adic numbers $\mathbb{Q}_p$) is infinite-dimensional. The sequence of unit vectors $e_1, e_2, \ldots$ defined as in the solution is linearly independent. The space $(\mathbb{F}_q)^\infty$ (infinite sequences with entries from the finite field $\mathbb{F}_q$) is infinite-dimensional. Again, the sequence of unit vectors $e_1, e_2, \ldots$ is linearly independent. The space of sequences $\ell^1(\mathbb{N})$ (infinite sequences $(x_n)$ with $\sum |x_n| < \infty$) over $\mathbb{R}$ or $\mathbb{C}$ is infinite-dimensional. The sequence of unit vectors $e_1, e_2, \ldots$ is linearly independent. The space of bounded sequences $\ell^\infty(\mathbb{N})$ (infinite sequences $(x_n)$ with $\sup |x_n| < \infty$) over $\mathbb{R}$ or $\mathbb{C}$ is infinite-dimensional. Once again, the sequence of unit vectors $e_1, e_2, \ldots$ is linearly independent. These examples demonstrate that the property of being infinite-dimensional is common among sequence spaces, regardless of the underlying field. The unit vectors always form a linearly independent set that can be extended indefinitely. This is true in the basic sequence space $\mathbb{F}^\infty$, in sequence spaces over other fields like $\mathbb{Q}_p$ and $\mathbb{F}_q$, and in sequence spaces with additional constraints like $\ell^1(\mathbb{N})$ and $\ell^\infty(\mathbb{N})$. --- $\textbf{Exercise 18.}$ Prove that $\mathbb{F}^\infty$ is infinite-dimensional. $\textbf{Solution 18.}$ For each positive integer $m$, let $e_m$ be the element of $\mathbb{F}^\infty$ whose $m$th coordinate equals 1 and whose other coordinates equal 0: $e_m = (0, \ldots, 0, \underbrace{1}_{\text{$m$th coordinate}}, 0, \ldots).$ Then $e_1, \ldots, e_m$ is a linearly independent list of vectors in $\mathbb{F}^\infty$, as is easy to verify. Exercise 17 now implies that $\mathbb{F}^\infty$ is infinite-dimensional. $\textit{Commentary:}$ This exercise asks to prove that a specific vector space, namely $\mathbb{F}^\infty$ (the space of infinite sequences with entries from $\mathbb{F}$), is infinite-dimensional. The proof uses the characterization of infinite-dimensional spaces from Exercise 17. It exhibits an infinite sequence of vectors in $\mathbb{F}^\infty$ with the property that any initial segment is linearly independent. The sequence used is the standard sequence of unit vectors $e_1, e_2, \ldots$, where $e_m$ has a 1 in the $m$th coordinate and 0s everywhere else. It's easy to see that any finite subset of these vectors is linearly independent, because a linear combination of them that equals zero must have all coefficients equal to zero (by looking at each coordinate separately). This proof is a standard way to show that sequence spaces like $\mathbb{F}^\infty$ are infinite-dimensional. The unit vectors form a natural linearly independent set that can be extended indefinitely. The exercise provides practice in applying the characterization of infinite-dimensional spaces from the previous exercise. It also reinforces the concept of the unit vectors and their linear independence. $\textit{Examples:}$ The space $(\mathbb{Q}_p)^\infty$ (infinite sequences with entries from the $p$-adic numbers $\mathbb{Q}_p$) is infinite-dimensional. The sequence of unit vectors $e_1, e_2, \ldots$ defined as in the solution is linearly independent. The space $(\mathbb{F}_q)^\infty$ (infinite sequences with entries from the finite field $\mathbb{F}_q$) is infinite-dimensional. Again, the sequence of unit vectors $e_1, e_2, \ldots$ is linearly independent. The space of sequences $\ell^1(\mathbb{N})$ (infinite sequences $(x_n)$ with $\sum |x_n| < \infty$) over $\mathbb{R}$ or $\mathbb{C}$ is infinite-dimensional. The sequence of unit vectors $e_1, e_2, \ldots$ is linearly independent. The space of bounded sequences $\ell^\infty(\mathbb{N})$ (infinite sequences $(x_n)$ with $\sup |x_n| < \infty$) over $\mathbb{R}$ or $\mathbb{C}$ is infinite-dimensional. Once again, the sequence of unit vectors $e_1, e_2, \ldots$ is linearly independent. These examples demonstrate that the property of being infinite-dimensional is common among sequence spaces, regardless of the underlying field. The unit vectors always form a linearly independent set that can be extended indefinitely. This is true in the basic sequence space $\mathbb{F}^\infty$, in sequence spaces over other fields like $\mathbb{Q}_p$ and $\mathbb{F}_q$, and in sequence spaces with additional constraints like $\ell^1(\mathbb{N})$ and $\ell^\infty(\mathbb{N})$.