# Is there an infinite dimensional vector space H with bounded linear operator from H onto H which preserves the inner product?

Let $H$ be the set of infinite convergent sequences of Real numbers, with the natural scalar multiplication, inner product and infinity norm.
Apart from the identity operator (which satisfies your requirements), the operator which transposes the first two elements of a sequence, is a bounded linear operator from $H$ to $H$, which is surjective and preserves the inner product.