On this wikipedia page https://en.wikipedia.org/wiki/Cone_(linear_algebra) , "a subset ##C## of a real vector space ##V## is a cone if and only if ##\lambda x## belongs to ##C## for any ##x## in ##C## and any positive scalar ##\lambda## of ##V##."(adsbygoogle = window.adsbygoogle || []).push({});

The book in this link https://books.google.com/books?id=P...does the cone contain the zero vector&f=false defines the cone as always containing the zero vector.

I am slightly perplexed, especially since a few other sources I have come across define it as having zero, or simply define ##\lambda \ge 0##, which implies that it contains the zero vector. So, my first question is, does the cone contain the zero vector? How is it typically defined?

My next question is, why does ##V## have to be a real vector space? Can't we have cones in ##\mathbb{C}^n## or ##M_n(\mathbb{C})##? In the wiki article, I see that they say the concept of a cone can be extended to those vector spaces whose scalar fields is a superset of the ones they mention. So, because "##\mathbb{R} \subseteq \mathbb{C}##," the very same definition of a cone can be extended to ##\mathbb{C}^n##, without any modification?

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# Definition of a cone

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