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A bijective linear map that preserves the norm exactly.
Let and be normed vector spaces. A surjective linear isometry is a surjective linear map satisfying
for every . Norm preservation makes injective, so it is a bijection and its inverse is also a linear isometry.
When , these maps form a group under composition and act on the unit sphere. Two normed spaces are linearly isometric when such a map exists between them.
A normed vector space is a vector space equipped with a norm on .
A linear map is a function between vector spaces over the same field such that for all and ,
For a normed vector space , its unit sphere is
It is the boundary of the closed unit ball . The term “sphere” refers to the given norm: unless the norm comes from an inner product, need not be round in Euclidean coordinates.
Every nonzero vector is a positive scalar multiple of a unique point of , namely .