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$F^\infty$ is a vector space defined as $\{(x_1, x_2...) : x_j \in F$ for $j = 1, 2, \ldots, n\}$. Is it correct?

A list can't be infinite in length, but an element of $F^\infty$ looks to be an $ \infty$-tuple.

Please, elaborate on this.

Prostitute
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  • It should be $j \in \Bbb N$. – Mark Fantini Jun 29 '14 at 21:00
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    Is what correct? That set may not be a vector space. There is no reason to think that there exists $0_{F^\infty}$. I like your nickname though. – Git Gud Jun 29 '14 at 21:01
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    A list certainly may be infinite in length. You seem to have defined a set that is frequently denoted $F^{\mathbb N}$, and this may be thought of as the set of all sequences of elements of $F$. No problem! And if $F$ is a field, the set is a vector space over $F$. – Lubin Jun 29 '14 at 21:02
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    This is from Axler's Linear Algebra textbook. He says "Sometimes we will use the word list without specifying its length. Remember, however, that by definition each list has a finite length that is a nonnegative integer, so that an object that looks like $(x_1,x_2, \ldots )$, which might be said to have infinite length, is not a list." The definition of the vector space above is also from his book. – Prostitute Jun 29 '14 at 21:07
  • Prostitute, as @Lubin said, if $F$ is field, then $F^\infty$ is a vector space. And obviously it is assumed that $F$ is a field. I missed that in my first comment. – Git Gud Jun 29 '14 at 21:14
  • In the case of $F=\mathbb{R}$, the notation $\mathbb{R}^\infty$ usually refers to the set of all sequences with only a finite number of nonzero elements, (at least in topology). – Peter Franek Jun 29 '14 at 21:45
  • It's important to say what the operations are. Or at least mention that they are the usual ones. – Ivo Terek Jun 29 '14 at 22:20
  • @GitGud Sorry but no: this user name is repulsive, and the justifications for it on another page even more so. – Did Jul 07 '14 at 08:09

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The symbolism $(x_1, x_2, \ldots)$ is not really rigorous. There is no such thing as a countable tuple. However the string of symbols $(x_1, x_2, \ldots)$ is supposed to make you recall $n$-tuples. An element of $F^\infty$ is actually a function from $\Bbb N$ to $F$. If you pick an element from $F^\infty$ which is written as $(x_1, x_2, x_3, \ldots)$ this is actually a string representing the images of the natural numbers. $1$ is sent to $x_1$; $2$ is sent to $x_2$; etc.

You can recast the definition of $F^n$ (which is usually defined as $F\times\cdots\times F$ $n$ times) as the set of all functions from the finite ordinal $n$ to $F$. This aligns with usage of $Y^X$ as the set of all functions from $X$ to $Y$.