How to say $\dim X=|X|$ without using mathematical symbols?

Question

I happen to be in a situation where I have to say $\dim X=|X|$ without using mathematical symbols. Here is the thing I am trying to say:

Let $X$ is a Banach space. $|X|$ denotes the cardinality of the space. If $\dim X=\infty$, then $\dim X=|X|$.

I try to say something like "The basis of an infinite-dimensional Banach space possesses the same cardinality as the entire space". This is a bit awkward.

Is there a better way of saying it?

Answer 1

1. "Any [Hamel] basis of an infinite dimensional Banach space has the same cardinality as the space itself", or

2. "There is a bijection between any [Hamel] basis of an infinite dimensional Banach space, and the entire space. I might add, this bijection is certainly not a linear transformation."

Those require you to know what is a Hamel basis and a Banach space, of course. Which may or may not be "as terrible" as mathematical symbols.