One generally takes the inner product to be positive-definite; otherwise, we say that is only a pseudo-Euclidean space. Also, one generally takes the dimension to be finite; Euclid himself only considered dimensions up to . For an infinite-dimensional Euclidean space, you would probably want to be a Hilbert space.
Arguably, the spaces studied by Euclid were not really modelled on inner product spaces, as the distances were lengths, not real numbers (which, if non-negative, are ratios of lengths). So we should say that has an inner product valued in some oriented line (or rather, in ). Of course, Euclid did not use the inner product (which takes negative values) directly, but today we can recover it from what Euclid did discuss: lengths (valued in ) and angles (dimensionless).
Since the days of René Descartes?, it is common to identify a Euclidean space with a Cartesian space, that is for the dimension. But Euclid's spaces had no coordinates; and in any case, what we do with them is still coordinate-independent.
Given two points and of a Euclidean space , their difference belongs to the vector space , where it has a norm
This real number (or properly, element of the line ) is the distance between and , or the length of the line segment . This distance function makes into an (-valued) metric space.
Given three points , with (so that ), we can form the ratio
which is a (dimensionless) real number. By the Cauchy–Schwartz inequality, this number lies between and , so it's the cosine? of a unique angle measure between and radians. This is the measure of the angle . In a -dimensional Euclidean space, we can interpret as a signed angle (so taking values anywhere on the unit circle?) if we fix an orientation of .
Conversely, knowing angles and lengths, we may recover the inner product on ;
and other inner products are recovered by linearity. (We must then use the axioms of Euclidean geometry to prove that this is well defined and actually an inner product.) It’s actually possible to recover the inner product and angles from lengths alone; this is discussed at Hilbert space.