Math CentralQuandaries & Queries


Question from Justin, a student:

Hello, I was just wondering are some infinite-dimensional spaces larger than others. Thanks a lot for your help with this question!

All the Best,


I'm assuming that you mean vector spaces here.

Yes, but (in the sense of cardinality at least) most of the ones you're likely to encounter will have the same size as the real line. The continuum hypothesis says that there may be uncountable numbers smaller than the cardinality of the reals, but as Cantor showed the reals are certainly uncountable. And transfinite cardinals have the property that αβ = α for 0<β<α. (On the other hand, αβ > β for α>1.)

The reals, finite dimensional real spaces, and countable-dimensional real spaces all have the same dimension as the reals. Even the space of continuous functions from the real line to itself has the same dimension, though it's a map out of an uncountable set, because it is determined by its values of a countable set (the rationals.)

To get to a vector space with bigger dimension you would have to look at something like the space of all functions R⇒R, continuous or not. This is not something we find very useful, especially in conjunction with the idea of dimension.

-Good Hunting!

Justin replied:

Hello again, I was also just wondering (in Hilbert Space and Function Space) are there infinite-dimensional spaces larger than each other in terms of cardinality? Thanks a lot for your help again!

All the Best,


Second Response:


There are sets of larger and larger (infinite) cardinalities. A "space" is just a set of points and some sort of operation or structure, for example the set of all functions from the set back to itself that satisfy some requirements. That suggests the answer is yes. I bet you can fill in enough details to make this more precise.


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