Name of Norm-preserving Linear Transformations?

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What is the common name for norm-preserving linear transformations in a normed linear space? I want to say they are the unitary transformations, but I'm just fuzzy enough not to know a good way of proving it.
 
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Linear isometries.
 
morphism said:
Linear isometries.

Thanks! This led me to wikipedia's page which had the equally helpful statement:

"The isometric linear maps from Cn to itself are the unitary matrices."

It's good to see my mathematical intuition is in tact after I've been thinking about physics lately =-D
 

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