The discussion revolves around the meanings and implications of the symbols $\mathbb{K}$ and $\mathbb{R}^n$ in mathematical contexts. Participants explore the definitions and assumptions associated with these symbols, particularly in relation to fields and vector spaces.
Participants express some agreement on the general meanings of $\mathbb{K}$ and $\mathbb{R}^n$, but there is disagreement regarding the implications and assumptions that should accompany these definitions, particularly concerning the vector space structure of $\mathbb{R}^n$.
Participants note that the definitions of $\mathbb{K}$ and $\mathbb{R}^n$ may depend on the context in which they are used, and that assumptions about vector space structures or algebraic properties may not always be explicitly stated.
A tiny disagreement. R^n is the set of order n-tuples of real numbers. It becomes a vector space only with the convention that "sum" and "scalar multiplication" are "coordinate wise". Yes, that is the "natural" convention but it separate from just "R^n".conquest said:the blackboard K is often used to refer to an arbitrary field. If you encounter it somewhere it should be stated what is meant by it.
The blackboard R^n just means n-dimensional real space. So basically (up to linear isomorphism) the unique n-dimensional vector space over the real numbers.