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)?

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

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?

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). This is why I resolved to edit the OP and drop the earlier assumption on the completeness of $\mathbf{X}$ and $\mathbf{Y}$.

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)?

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