What is the difference between a Hamel basis and a Schauder basis?

Solution 1:

People keep mentioning the restriction on the size of a Schauder basis, but I think it's more important to emphasize that these bases are bases with respect to different spans.

For an ordinary vector space, only finite linear combinations are defined, and you can't hope for anything more. (Let's call these Hamel combinations.) In this context, you can talk about minimal sets whose Hamel combinations generate a vector space.

When your vector space has a good enough topology, you can define countable linear combinations (which we'll call Schauder combinations) and talk about sets whose Schauder combinations generate the vector space.

If you take a Schauder basis, you can still use it as a Hamel basis and look at its collection of Hamel combinations, and you should see its Schauder-span will normally be strictly larger than its Hamel-span.

This also raises the question of linear independence: when there are two types of span, you now have two types of linear independence conditions. In principle, Schauder-independence is stronger because it implies Hamel-independence of a set of basis elements.

Finally, let me swing back around to the question of the cardinality of the basis. I don't actually think (/know) that it's absolutely necessary to have infinitely many elements in a Schauder basis. In the case where you allow finite Schauder bases, you don't actually need infinite linear combinations, and the Schauder and Hamel bases coincide. But definitely there is a difference in the infinite dimensional cases. In that sense, using the modifier "Schauder" actually becomes useful, so maybe that is why some people are convinced Schauder bases might be infinite.

And now about the limit on Schauder bases only being countable. Certainly given any space where countable sums converge, you can take a set of whatever cardinality and still consider its Schauder span (just like you could also consider its Hamel span). I know that the case of a separable space is especially useful and popular, and necessitates a countable basis, so that is probably why people tend to think of Schauder bases as countable. But I had thought uncountable Schauder bases were also used for inseparable Hilbert spaces.

Solution 2:

The problem is that an element of a Hamel basis might be an infinite linear combination of the other basis elements. Essentially, linear dependence changes definition.

Solution 3:

Maybe a good point to start is this useful corollary of Baire Cathegory Theorem

the cardinality of an Hamel base of a Banach Space can be finite or uncountable. It can't be countable

The proof is a delightful application of Baire theorem.

Now to give an explicit example, we can consider the space $\ell^2 $ which has the standard base $ M:=$ $\lbrace e_n \rbrace $ which is not an Hamel base, but an Hilbert base (or Schauder, in this case the two coincide). To see the differences consider the linear span of $ M $. It's trivial to see that it is $ c_{00} $ but (using orthonormality property of $ M $) each vector $ v \in \ell^2 $ can be expressed as $ v=\sum_{k=1}^{\infty}(v, e_k) e_k $

In fact the restriction to FINITE linear combinations is a strong restriction. Let me show you another similar example. Consider $ c_0 $ the Banach space of the sequences convergent to 0. $ M$ is a Schauder base of it (verify it) but given for example $ u= \lbrace \frac {1}{n}_n \rbrace $ you can't express u as a finite linear combination of elements of M . So changing the meaning of the base in fact change "how big is its span"

Solution 4:

In the case of an infinite dimensional complete space, if you have a Banach space, then any Hamel basis is not countable. On the other hand, any Schauder basis has to be countable.