Normed Vector Space
In a normed vector space V, one of the defining properties of the norm is the triangle inequality:
that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
If the normed space is euclidean, or, more generally, strictly convex, then if and only if the triangle formed by ,, and, is degenerate, that is, and are on the same ray, i.e., or, or for some . This property characterizes strictly convex normed spaces such as the spaces . However, there are normed spaces in which this is not true. For instance, consider the plane with the norm (the Manhattan distance) and denote and . Then the triangle formed by ,, and, is non-degenerate but
Read more about this topic: Triangle Inequality
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“No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.”
—Isaac Newton (1642–1727)