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We characterize finite codimensional linear isometries on two spaces, C(n)[0; 1] and Lip [0; 1], where C(n)[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C(n)[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.

Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions