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Let \(U \in \text{End}(V)\). Then \(U\) is unitary if and only if \(\left\| Uv \right\| = \left\| v \right\|\).
Let \(U \in \text{End}(V)\). Then \(U\) is unitary if and only if \(\left\| Uv \right\| = \left\| v \right\|\).