We could define the following notion of basis in a way analogous to unconditional Schauder basis:

If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. We say that $B$ is a basis if, for every $x\in X$ there exists a unique "coefficient function" $c\colon I\to\mathbb R$ such that $$\sum_{i\in I} c(i) b_i =x.$$

Probably more natural notation -- resembling the usual notation for linear combinations -- would be $\sum\limits_{i\in I} c_i b_i =x$.

The sum $\sum\limits_{i\in I} v_i$ of elements of a topological vector space is defined as a limit of the net $$x_F=\sum_{i\in F} x_i$$ on the directed set consisting of all finite subsets of $I$ (ordered by inclusion).

This seems to be the usual approach for defining sums over an index set which is not necessarily countable. It is defined in this way, for example, in this answer.

  • Does this type of basis have a name?
  • Are such basis studied? Are they useful in some areas of mathematics?

For example, orthonormal basis (maximal orthonormal set) in a Hilbert space is such basis.

Motivation for this questions is that it seems to be a very natural next step after defining Hamel basis and Schauder basis.

In a typical curriculum, most people usually encounter the word basis for the first time in connection with vector space. If we only have structure of a vector space, we only can do finite sums and finite linear combinations. So trying to define basis using something like $\sum\limits_{i=1}^\infty c_i b_i$ is not possible.

But later we learn about normed spaces and topological vector space, where we have the notion of convergence and thus the expression $\sum\limits_{i=1}^\infty c_i b_i$ makes sense. So we now can define the Schauder basis and the definition seems very similar to the definition of Hamel basis - we just replaced finite sum by an infinite series.

Now we could try to define the analogous notion for arbitrary index set, not just $\mathbb N$. One possible way to do this would be to use the notion of sum over arbitrary index set which I described above. (On $\mathbb N$ we have a natural ordering, while on arbitrary set $I$ we do not have an order which we could call "natural". Therefore this notion is closer to unconditional Schauder basis than to Schauder basis.)

When trying to find some references for this type of basis I found the following remark in Heil's Basis Theory Primer (doi:10.1007/978-0-8176-4687-5; some version of the manuscript seems to be available on the author's website:

Remark 4.4: (d) The definition of basis requires that $\{x_n\}$ be a countable sequence. Sometimes, as in Exercises 3.6 and 3.7, it is possible to deal with uncountable systems that have basis-like properties, but to avoid confusion we will not call such systems bases.

The above remark suggest that the type of basis I described above is probably not very useful. (By the way the Exercise 3.6 and 3.7 mentioned in this remark contain, among other things, the fact that orthonormal basis of a Hilbert space has such properties.)

On the other hand, special case of such basis is an orthonormal basis in a Hilbert space. I'd say that the fact that an inner-product space has an orthonormal basis gives some useful information about this space. But I guess that it might be difficult to use similar basis in a sensible way in a Banach space, which is not a Hilbert space.

Despite this skepticism, I posted the question here. It is still possible that I will learn from answers about some unexpected applications.

EDIT: Almost immediately after posting this question I have noticed this older question: What do we call a Schauder-like basis that is uncountable?

I would argue that the questions are a bit different. The other posts only asks about the name for such basis, not for applications. And the OP does not describe how exactly they want to go from the countable case to the arbitrary case. (Although to me the above seems the most natural way, the phrasing of the other question leaves also other possibilities open. (And in fact, a comment posted there describes some kind of basis which seems different from my definition above.)

I will leave it to other users to judge whether the two questions are different enough to remain open, or whether one of them should be closed as a duplicate.

EDIT 2: After a bit of searching I found a post on MO which asks (if I understand it correctly) whether the cardinality of this type of basis (assuming it exists) is determined uniquely by the space $X$: Uniqueness of dimension in Banach spaces

  • $\begingroup$ Note that if $X$ is first-countable (in particular, if it's a Banach space), if you have such a basis, then for any $x\in X$, the coefficient function $c$ has countable support, like in the Hilbert space case. $\endgroup$ – Eric Wofsey Nov 7 '15 at 23:09
  • $\begingroup$ "On $\mathbb{N}$ we do not have a natural ordering, ..." ? $\;$ $\endgroup$ – user57159 Nov 8 '15 at 18:11
  • $\begingroup$ That was, naturally, a typo. Thanks for letting me know @RickyDemer, it should be corrected now. $\endgroup$ – Martin Sleziak Nov 8 '15 at 18:17
  • $\begingroup$ @EricWofsey : $\;\;\;$ I believe one only gets that the support of the coefficient function is a countable $\hspace{.38 in}$ union of finite sets. $\:$ (Going the rest of the way is then equivalent to Countable Choice for Finite Sets.) $\hspace{.29 in}$ $\endgroup$ – user57159 Nov 8 '15 at 18:27

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