are ellipses allowed when forming rules like this?

>>8938215
yes

It's a convenient shorthand for notating an arbitrary vector in [math]\mathbb{R}^n[/math].

If that seriously triggers you, you can treat [math](x_1, \cdots, x_n)[/math] as shorthand for a function [math]x:\{1, \cdots, n\} \to\mathbb{R}[/math] such that [math]i \mapsto x_i[/math]. Then the vector space [math] \mathbb{R}^n[/math] is simply the set of all functions with domain [math]\{1, \cdots, n\}[/math] and codomain [math]\mathbb{R}[/math].

(Personally I prefer this definition, since it generalizes nicely to infinite-dimensional vector spaces. It also makes a clear distinction between a vector space and its dual.)

>>8938226
>generalizes nicely to infinite-dimensional vector spaces.
how does this work past the first infinite cardinality?

>>8938215
>are ellipses allowed when forming rules like this?

Yes because n is finite.

>>8938232
Hilbert spaces of functions from R to R

>>8938232
It's as simple as swapping out {1,...,n} with the cardinal of your choice, then pulling back all vector operations on the codomain [math]\mathbb{R}[/math].

You might argue that you're not really interested in the set of "all" R-valued functions, but only the continuous ones. Which can be done, but the process is slightly more complicated: basically you'll need the domain to be an ordinal instead of a cardinal, and then require that the vector preserves the order structure thus induced. In practice most people just invoke the axiom of choice which handwaves away the distinction by allowing you to use cardinals as if they were ordinals.

>>8938215
Since they're comma separated then they should be \ldots (dots along the bottom) not \cdots (dots along the center).

They are allowed because the quantity is finite and there is no ambiguity as to how many elements are enumerated.

The only time ellipses aren't allowed is when they create some ambiguity and it matters in some way. For instance you can't say
>0.000...01
to refer to some "infinitesimal" quantity.

>>8938256
>In practice most people just invoke the axiom of choice which handwaves away the distinction by allowing you to use cardinals as if they were ordinals.

(Spoiler: They are)
(Spoiler explanation: The axiom of choice is unjustifiable and uncountable infinities don't exist you fucking mongoloid)

>>8938256
It only generalizes if you accept the axiom of choice anyway, since the existence of a basis in any vector space is equivalent to it.
And Hammel basis are practically useless in analysis, and, for exemple, in Banach spaces you're interested in Schauder basis and not all of them have one. The concept of basis generalizes well only on Hilbert spaces.

Reminder that the notion of ''dimension'' or ''cardinality'' is already useless in predicative mathematics, because a finite set can have infinite subsets

>>8938509
>because a finite set can have infinite subsets
What?

>>8938509
Reminder that material set theory is useless for doing linear algebra (or any algebra at all) and anyone defining mathematical structures in terms of set membership is peddling unfalsifiable woo as truth.

>>8938215

Why do they bother making definitions for R^N like that?

Does not

[math]\mathbb{R^2 = R \times R}[/math]

?

File: Random_cfaa38_6189452.jpg (16KB, 446x359px) Image search: [Google]

16KB, 446x359px

>>8938509

>>8938342
Numbers don't exist you brainlet.

>>8939187
what? they aren't making up any definitions or anything, they are simply stating that the algrebraic rules for R^N are what you would expect given the rules in R2, R3 etc -- the same principles extend to R^N

they aren't making anything up or saying here this is how we define R^N, just confirming that algebra still works lol

You gonna stop them fuccboi?