Bonjour/bonsoir à toutes et à tous.

This may really be a very basic question, but... Let $\mathbf{X} \equiv (X, \|\cdot\|_X)$ and $\mathbf{Y} \equiv (Y, \|\cdot\|_Y)$ be surjectively isometric Banach spaces (over the real or the complex field).

> **Question 1.** Do $\mathbf{X}$ and $\mathbf{Y}$ need to share the same Hamel
> dimension?

The answer is clearly yes if the scalar field is the real numbers, since then any surjective isometry $\mathbf{X} \to \mathbf{Y}$ is, a fortiori, an affine transformation $X \to Y$ (via the Mazur-Ulam theorem). But what about the complex case?

> **Question 2.** Do $\mathbf{X}$ and $\mathbf{Y}$ need to share the same (extended) Schauder
> dimension as defined in J. W. Evans
> and R. A. Tapia, *Hamel Versus Schauder
> Dimension*, The American Mathematical
> Monthly, Vol. 77, No. 4 (Apr., 1970),
> pp. 385-388 ([click][1])?

**Notes.** 1) I'm using the term *isometry* to refer, herein, to both linear and non-linear isometries.


  [1]: http://www.jstor.org/stable/2316148