Vector space/Inner product/Isometry/Functorial properties/Exercise

Let ${}U,V,W$ be ${}{\mathbb {K} }$-vector spaces, each endowed with an inner product. Show the following statements.

a) The identity ${}V\rightarrow V$ is an isometry.

b) If ${}\varphi \colon U\rightarrow V$ is a bijective isometry, then the inverse mapping ${}\varphi ^{-1}$ is also an isometry.

c) If ${}\varphi \colon U\rightarrow V$ and ${}\psi \colon V\rightarrow W$ are isometries, then the composition ${}\psi \circ \varphi$ is also an isometry.

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