The relationship among different types of fundamental spaces.

Question

I'm just looking to make sure my understanding of certain fundamental spaces are correct.

Denote the set of all vector spaces by $V$, the set of all metric spaces by $M$, the set of all normed spaces by $N$, the set of all Banach spaces by $B$, and the set of all Hilbert spaces by $H$. Then I believe the following is true:

$$H \subset B \subset N \subset (V \cap M)$$

Is this understanding correct? If not, this seems that way it is in the beginning of functional analysis, so is this like the generic situation (or introduction)?

Answer 1

The idea is correct. The language is not, since there is no such thing as the set of all vector spaces, the set of all metric spaces, and so on. But, yes,

• every Hilbert space is a Banach space;
• every Banach space is a normed vector space;
• every normed vector space is both a vector space and a metric space.