$\mathbb{K}$ (field)

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Created to define what $\mathbb{K}$ means when encountered as a field (eg if $(X,\mathbb{K})$ is a vector space - Alec (talk) 03:54, 8 April 2017 (UTC)

• Also make sure that the doctrine page lists references that this page does, as they're the same thing!

Meaning

As per the Doctrine:K (field) we use $\mathbb{K}$ to denote a field which is either the field of real numbers, $\mathbb{R}$, or the field of complex numbers, $\mathbb{C}$.

This only applies if we do not explicitly define what $\mathbb{K}$ is, in the absence of any mention of what $\mathbb{K}$ is the doctrine applies and it is a stand in for either $\mathbb{R}$ or $\mathbb{C}$.

This convention is not unusual.

Caveats

1. If we say "let $(X,\mathbb{K})$ and $(Y,\mathbb{K})$ be vector spaces" it is unclear whether $X$ and $Y$ are both over the same field or not.
   • We will qualify this with "over the same field" or "over potentially distinct fields"; or use $\mathbb{K}_1$ and $\mathbb{K}_2$ if the fields will play a part in what is to come.
2. If we say "let $(X,\mathbb{K})$ be a vector space over the field $\mathbb{K}$" it suggests that $\mathbb{K}$ is just being used to represent the field, the doctrine doesn't apply then. It'd be as if the more conventional $\mathbb{F}$ were used instead.