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 (1) *normed* 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?

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**Added later.** As pointed out in the comments below, the answer to Question 1 is still yes if $\mathbf{X}$ and $\mathbf{Y}$ are (real or complex) Banach spaces essentially because the Hamel dimension of an infinite-dimensional (real or complex) Banach space is always at least the cardinality of the continuum (even if the CH fails) as proved in H. E. Lacey, *The Hamel Dimension of any Infinite Dimensional Separable Banach Space is c*, The American Mathematical Monthly, Vol. 80 (1973), p. 298 ([click][1]). This is why I resolved to edit the OP and drop the earlier assumption on the completeness of $\mathbf{X}$ and $\mathbf{Y}$.

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> **Question 2.** Now assuming that $\mathbf{X}$ and $\mathbf{Y}$ are Banach, do they 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][2])?

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


  [1]: http://www.jstor.org/stable/2318458
  [2]: http://www.jstor.org/stable/2316148