Transport of structure

In mathematics, transport of structure is the definition of a new structure on an object by reference to another object on which a similar structure already exists. Definitions by transport of structure are regarded as canonical.

Since mathematical structures are often defined in reference to an underlying space, many examples of transport of structure involve spaces and mappings between them. For example, if V and W are vector spaces, and if ${\displaystyle \phi \colon V\to W}$ is an isomorphism, and if ${\displaystyle (\cdot ,\cdot )}$ is an inner product on ${\displaystyle W}$, then we can define an inner product ${\displaystyle [\cdot ,\cdot ]}$ on V by

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