Sentence:  Let K be a field with discrete valuation v, and \mathcal{O}_K be the valuation ring. / Definition 2.3.1 A <WSD>"p"-adic norm</WSD> on vector space V over K is a function \alpha:V\rightarrow\R satisfying: / a) \alpha(x)\ge 0 and \alpha(x)=0 if and only if x=0. / b) \alpha(ax)=\vert a\vert \alpha(x) for a\in K and x\in V. / c) \alpha(x+y)\le\operatorname{sup}(\alpha(x),\alpha(y)) for x,y\in V. / If \alpha is a <WSD>"p"-adic norm</WSD> and t>0, then the dilation t\alpha is a <WSD>"p"-adic norm</WSD>, and we denote by N_p(V) the set of dilation classes of <WSD>"p"-adic norms</WSD> on V. 
Meaning of the word marked with <WSD> tags:  A norm on a vector space which is defined over a field equipped with a discrete valuation (a generalisation of "p"-adic absolute value).