Wikipedia: History of Normed vector space

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Changed: 13c13

Categorically speaking, a homomorphism of normed vector spaces would be a linear map that preserves the norm. This isn't very useful, so a notion which may be more appropriate to [topological vector spaces]? is often used: a homomorphism is a linear map that is continuous. When referring to a norm-preserving linear map, the term isometry is used. Note that an isometry is automatically an isomorphism (its inverse is an isometry as well.) When speaking of isomorphisms of normed spaces, one normally means an isometry, or at the very least a continuous, onto linear map with a continuous inverse.

Categorically speaking, a morphism of normed vector spaces would be a linear map that preserves the norm. This isn't very useful, so a notion which may be more appropriate to [topological vector spaces]? is often used: a morphism is a linear map that is continuous. When referring to a norm-preserving linear map, the term isometry is used. Note that an isometry is automatically an isomorphism (its inverse is an isometry as well.) When speaking of isomorphisms of normed spaces, one normally means an isometry, or at the very least a continuous, bijective linear map with a continuous inverse.

	Revision 5 . . September 30, 2001 5:53 am by AxelBoldt *[links]**
	Revision 4 . . (edit) September 25, 2001 12:40 pm by AxelBoldt
	Revision 3 . . August 16, 2001 9:04 pm by Zundark [import some text from Linear_Algebra/Normed_Vector_Space]