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I'd searched them for a while, but still have not found a clear and unity definition on it. The problem really confused me.

What is the precise definition of Cartesian coordinate and Euclidean space?

Is the definition of Cartesian coordinate including some non-abstract(in a geometric manner?) notion, such like axes, origin and the specify of direction? Or it is just a purely set, such like $\mathbb{R}^2$ or $\mathbb{R}^3$, possibly with the operations defines on it?

The same questions comes to the definition of Euclidean spaces. What does it mean? Does it include some non-abstract(in a geometric manner?) notion, such like axes, origin and the specify of direction? Is it different to Cartesian space? Or, is any vectors space $V$ over any field(including $\mathbb{C}$) that has an inner product $<\cdot,\cdot>$ defines on it can altogether be called a Euclidean space?

And the last of all, what is a real coordinate system?

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  • $\begingroup$ Concerning the definition of Euclidean spaces: An Euclidean space is a normed vector space where the norm is the Euclidean norm. The Euclidean norm is the so called 2-norm $\|\mathbf{x}\|_2:=\sqrt{x_1^2+\ldots+x_n^2}$. $\endgroup$
    – Xaver
    May 28, 2016 at 15:51
  • $\begingroup$ Concerning the definition of Cartesian coordinate systems I am not completely sure. But in my experience, coordinate systems that are orthogonal are called Cartesian (in contrast to other systems such as polar coordinate systems). $\endgroup$
    – Xaver
    May 28, 2016 at 16:01
  • $\begingroup$ The term "real" in "real coordinate system" means that all coordinates are real numbers. $\endgroup$
    – Xaver
    May 28, 2016 at 16:02

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