A normed vector space is a pair (V, ‖·‖ ) where V is a vector space and ‖·‖ a norm on V.
A seminormed vector space is a pair (V,p) where V is a vector space and p a seminorm on V.
We often omit p or ‖·‖ and just write V for a space if it is clear from the context what (semi) norm we are using.
In a more general sense, a vector norm can be taken to be any real-valued function that satisfies these three properties. The properties 1. and 2. together imply that
- if and only if .
A useful variation of the triangle inequality is
- for any vectors x and y.
This also shows that a vector norm is a continuous function.
Read more about Normed Vector Space: Topological Structure, Linear Maps and Dual Spaces, Normed Spaces As Quotient Spaces of Seminormed Spaces, Finite Product Spaces
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“The merit of those who fill a space in the world’s history, who are borne forward, as it were, by the weight of thousands whom they lead, shed a perfume less sweet than do the sacrifices of private virtue.”
—Ralph Waldo Emerson (1803–1882)