We investigate linear isometries on various Banach and Fréchet spaces. Stephan Banach initiated this study on the space $X$ of continuous functions on a compact set endowed with the supremum norm. He proved in 1930s that linear surjective isometries are weighted composition operators. Forelli, in 1964, obtained, without the hypothesis of sujectivity, a description of linear isometries on Hardy spaces $H^p(\mathbb{D})$, $p\neq 2$, $1\leq p<\infty$ as weighted composition operators where he replaced a function that Banach required to simply be a homeomorphism with an inner function. In this course we describe and give examples of inner functions as described by Beurling in 1954 who proved that every closed invariant subspace of the unilateral shift other than {0} has the form $\phi H^2$ where $\phi$ is an inner function. We share a recent result of 2025 by Chalendar, Oger and Partington on linear isometries on the Fréchet space $Hol(\mathbb{D}).$ We conclude the course with a few open questions.