The standard norm on a Hilbert space is induced from the inner product. However if there is another norm on the Hilbert space that is equivalent to the standard norm, is this equivalent norm necessarily induced from an inner product?
The equivalent norm gives the same topology as the standard one, but I am not sure about the inner product itself.
The question was answered in the comments. A norm need not come from an inner product.
However, there is something positive to be said. If a norm comes from an inner product it satisfies an identity called the parallelogram law. If a norm satisfies the parallelogram law you can actually define the inner product from the norm. More information on the Wiki page.