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A linear isometry is a map $F: V \to W$ between two inner product spaces that is both a vector space isomorphism and preserves inner products. This means it maintains the geometric structure defined by the inner product.

Conditions: $V$ and $W$ are inner product spaces.; $F$ is a vector space isomorphism.; $F$ preserves inner products for all $a, b \in V$.

A linear isometry is a map $F: V \to W$ between two inner product spaces that is both a vector space isomorphism and preserves inner products. This means it maintains the geometric structure defined by the inner product.

Conditions: $V$ and $W$ are inner product spaces.; $F$ is a vector space isomorphism.; $F$ preserves inner products for all $a, b \in V$.