isometry

An isometry is a map that preserves distance: ||T u - T v|| = ||u - v|| for all u, v. For linear maps this is the same as preserving norm, ||T v|| = ||v||, since T(u - v) = T u - T v. It moves the space without stretching, shrinking, or tearing — only rotating and reflecting.

Algebraically a linear isometry is exactly T^* T = I. This makes T injective (it has a left inverse) and forces its columns to be orthonormal. By polarization, preserving norms automatically preserves the full inner product, so isometries are angle-preserving too, not just length-preserving.

The crucial subtlety separating isometry from unitary is surjectivity. T^* T = I gives a left inverse but not necessarily a right one. In finite dimensions a linear injection from V to V is automatically onto, so a self-map isometry IS unitary. The distinction only bites between different spaces or in infinite dimensions.

The classic infinite-dimensional counterexample is the right shift on sequences, (x_1, x_2, ...) -> (0, x_1, x_2, ...). It preserves every norm (an isometry, T^* T = I) yet misses everything with a nonzero first coordinate, so it is not onto and T T^* != I. An isometry that fails to be unitary is the hallmark of infinite dimensions.