Understanding $\mathbb{K}$ and $\mathbb{R}^n$ - Get Your Answers Here

[Hernaner28]

Hi. What does that mean when we see:

$$\mathbb{K}$$ what set is that? definitely not the reals, integers, etc.
and
$$\mathbb{R}^n$$ Is that the reals or what?

Thanks!

 

[conquest]

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.

 

[Hernaner28]

Thank you!

 

[HallsofIvy]

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.

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]

You are right of course. Naturally my response was in terms of the vector space since this is the first definition I saw. You could also assume this and express that you are only talking strictly as the n-tuples of real numbers by mentioning you use the bare set underlying the vector space. I think in most literature the vector space (indeed also the natural topological, norm, inner product, Lie group and manifold (symplectic, smooth, Riemannian)) structure are assumed when used. A thing that might be stated explicitely is any algebra structure.

So immediately when I see that symbol these things I also assume, but it might pay to be a bit more reserved about this.