Measures of non-compactness are most commonly used if M is a normed vector space.

In a normed vector space there is only one vector of norm equal to 0.

In the case of a normed vector space, the statement is:

It is not specifically the inner product on a normed vector space.

A vector space with a norm is called a normed vector space.

The open and closed balls in a normed vector space are balanced sets.

For this reason, not every scalar product space is a normed vector space.

This is the standard topology on any normed vector space.

He was one of the first mathematicians to apply normed vector spaces in numerical analysis.

The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.