I have this question in reviewing for my exam.
Let $H$ be an infinite dimensional Hilbert space. Write down an inner product on $H$ that gives a norm inequivalent with the original norm. Is $H$ complete under the norm determined by the new inner product?
In my understanding, as the norm of a Hilbert space is induced by the inner product it equipped, there are different inner products, all satisfy linearity, conjugate symmetry, positive-definiteness, hence different norms. My confusion is, why the question is asking "original norm"? Is a Hilbert space uniquely determined by the inner product it uses? For example, the collection of square integrable functions on $R^d$ is a Hilbert space and equipped with a inner product of integral form. Finite-dimensional complex Euclidean space with dot product. Or, there is a space can be equipped with different inner product (to define its norm) to form the same Hilbert space? For example, the Euclidean space?
BTW, how is this question related to the fact that.
All infinite-dimensional (separable) Hilbert spaces are $l^2$$(Z)$ in disguise
Due to my poor understanding of functional analysis, I am not sure how is this question trying to motivate my thinking. Any help would be appreciated! Thanks in advance!