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Why would it be incorrect to describe to the vectors of an infinite dimensional vector space as "infinite tuples"?

As far as I know, a tuple must be finite. But the objects of vector spaces are called vectors. Vectors are, generally, satisfactorily represented by tuples. In the case of an infinite vector space, however, it appears that tuples cannot be used without introducing some notion of an "infinite tuple." Why do tuples need to be finite (when a sequence can be infinite)?

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  • $\begingroup$ Who says that doing so would be incorrect? $\endgroup$ – Omnomnomnom May 31 '18 at 1:50
  • $\begingroup$ I can't find "infinite tuples" anywhere and I'm confused as to why tuples are defined as finite sets (not just collections but sets) with an order to begin with. Similarly, people often seem to describe tuples as collections of objects with an order, but if a tuple isn't a set, not just a collection, then a vector space can contain the same vector twice. I assume this isn't true? $\endgroup$ – asdfdsas May 31 '18 at 1:57
  • $\begingroup$ In fact a sequence can be represented by an infinite-tuple. But could you imagine elements of a vector space where each vector has a "number" of coordinates bigger than the cardinality of the natural numbers? $\endgroup$ – Bernard Massé May 31 '18 at 1:58
  • $\begingroup$ My understanding was that a sequence has to contain all of the same type/have some common characteristic, whereas a tuple did not have this restriction. I can't imagine such a space and yet I'm aware that infinite vector spaces exist. Are you saying that I can't "write down" a basis for such a space, so using an infinite-tuple to represent it is meaningless..? $\endgroup$ – asdfdsas May 31 '18 at 2:02
  • $\begingroup$ Even in finite vector spaces, the objects in the tuple are generally elements of the same underlying field $\endgroup$ – Omnomnomnom May 31 '18 at 2:17
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Some vector spaces are represented with infinituples: in particular, sequence spaces. If you think about it, tuples from the same set can be thought of as functions from finite sets $\lbrace 1, 2, \ldots, n \rbrace$ to the set. If you want an infinite version of this, just make the set into $\mathbb{N}$. A function from $\mathbb{N}$ to a set is a sequence!

But, for many vector spaces, this infinituple idea is impossible. It all comes down to bases. In finite dimensions, we have one dominant kind of basis: the Hamel basis. A Hamel basis is a set of vectors $B$ with linear independence and spanning. Linear independence means that a null linear combination of any finite subset of $B$ must be the trivial linear combination. Spanning means that every element of the space can be expressed as a linear combination of a finite number of elements of $B$.

Note that this definition varies slightly from the one given in finite-dimensional linear algebra courses, in that it allows for infinite sets $B$, and compensates by only considering finite subsets. However, we still get a uniqueness of representation, allowing for, in a sense, coordinate vectors!

Now, it can be shown with Zorn's lemma that every vector space has a Hamel basis. Unfortunately, in infinite dimensions, it is typically an intractable problem to construct such sets, let alone do anything with their representation.

Infinite dimensional vector spaces are often considered with norms (or at least locally convex topologies), giving them a topological bent as well. Hamel bases pay no respect to topology, and consequently are rarely used. Schauder bases are more common, as they allow for infinite sums (which require convergence, and hence topology).

These infinite sums allow for countable representations, where a Hamel Basis will often become uncountable. That's another problem with these basis representations: they can easily become uncountable, and thus not really possible to represent as an infinite list of field elements. In fact, many interesting infinite dimensional spaces are Banach spaces, and infinite-dimensional Banach spaces always have uncountable Hamel bases (it's a nice exercise in the Baire Category Theorem).

Basically, to sum up:

  • There are multiple types of bases in infinite dimensions.
  • Bases can be uncountable.
  • Bases can also be very difficult to construct/work with.

There are prominent examples of infinite representations (e.g. Fourier series) being useful, but typically these rely on countability and the underlying topology. There is also a whole branch of functional analysis devoted to looking at various bases!

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