Question

Question #65104

Answer 1

Yes. Both of them are normed linear spaces which are complete in the sense that every Cauchy sequence in the space converges to a limit point within the space.

However, the norm of a vector in a Banach space is not necessarily defined from the inner product. (Even though inner product is defined on the space)

On the other hand, for any vector `|f>` in a Hilbert space `V`, the norm may be defined as,

`||f|| = sqrt (< f|f>)` where `< f|f>` denotes an inner product of `|f>` (from `V`) with `< f|`
(from the dual space `bar V`).

Thus norms in the Hilbert space are defined using the inner product.

This may not be true for a Banach space.