Are all bounded linear operators including the ones that are Banach, also isometries?

Question

Are all bounded linear operators including Banach bounded linear operators, also isometries?

An isometry is a homeomorphism that preserves distance, i.e. only reorders the points.

May an unbounded linear operators, including those Banach and not banach be an isometry?

Answer 1

There's plenty of bounded linear operators which are not isometries, such as scaling by a constant (different from $\pm 1$).

Unbounded operators cannot be isometries, since isometries are continuous while unbounded operators aren't.