# Example 1.22 Axler's Linear Algebra Done Right notation confusion

I'm self-teaching my way through linear algebra done right by Axler, I'm using the 3rd edition. However, I'm a bit confused by the wording in Example 1.22:

$$\mathbf{F}^{\infty}$$ is defined to be the set of all sequences of elements of $$\mathbf{F}$$: $$\mathbf{F}^{\infty} = \{ (x_1,x_2,...):x_{j}\in \mathbf{F} \text{ for } j = 1,2,...\}$$

Following from definition in the first part of chapter 1, I know that $$\mathbf{F}^{n}$$ denotes any list of length n with elements of $$\mathbf{F}$$.

• so the notation here would suggest $$\mathbf{F}^{\infty}$$ is any list of infinite length

• but the words suggest that it's something like any possible list of arbitrary length

• or the set of all possible lists with members of F and arbitrary length $$\{ (x_1),\;(x_2),\;(x_1,x_2),\;(x_3),\;(x_1,x_3),\;(x_2,x_3),\; (x_1,x_2,x_3), ...\}$$

• or the set of all lists of infinite length and members of $$\mathbf{F}$$

or something else?

• $F^\infty$ is a set. The elements of this set are infinite lists, where each element of the list is an element of $F$. I think you’re confused because you think a “sequence” can be a finite list. That’s not true. A finite list of length $n$ is called an $n$-tuple, not a sequence. A sequence, by contrast, is always infinite in length. Jan 14, 2022 at 2:28

The elements of $$\mathbb{F}^\infty$$ are sequences (lists of infinite length), where every element in the sequence belongs to the field $$\mathbb{F}$$. For example, $$(1, 1/2, 1/3, 1/4, \ldots ) \in \mathbb{R}^\infty$$, but $$(1,2) \not\in \mathbb{R}^\infty$$.