Hilbert Space and Banach Space

Asked Dec 23 '19 at 00:42

Active Dec 23 '19 at 00:42

Viewed 75 times

Both are vector spaces
Both are normed vector spaces.
Both are complete

The only extra feature of a Hilbert space is an inner product which induces a norm.

We also know that for a norm which satisfies the parallelogram law, there exists an inner product which induces that norm. (correct? )

Suppose a Banach space has a norm which satisfies the parallelogram law. But we can't conclude that this space is Hilbert, because we have to explicitly define an inner product, right?

So the following definition is wrong: A Hilbert space is a Banach space with a norm that satisfies the parallelogram law.

The purpose of this question is just to get slight clarification on terminology. That's all.

asked Dec 23 '19 at 00:42

rims

2,657

4

if a norm satisfies the parallelogram law, it is induced by an inner product defined by the polarization identity – J. W. Tanner Dec 23 '19 at 00:48
1

Related (your suggested definition appears in the first comment) – Pedro Dec 23 '19 at 01:15

Hilbert Space and Banach Space

0 Answers0