Measure of non-compactness

Question

Can someone give me some simple examples of measure of non-compactness of sets in Banach spaces or metric spaces, which are easy to understand.

Answer 1

Sets: For a bounded subset $A$ of a Banach space $X$ you can define $$ \gamma(A) = \inf\{\epsilon > 0 : A \text{ can be covered by finitely many balls of radius } \epsilon \}$$ and you have $\gamma(A) = 0$ iff $A$ is relatively compact.

Operators: Let $B(X)$ the Banach space of bounded operators acting on $X$ and $T\in B(X)$.

• Using the above definition for sets you can set $\gamma(T) = \gamma(T B_X)$ where $B_X$ is the unit ball in $X$.
• Define $$\|T\|_\mu = \inf \{\|T|_M\|:M\subset X \text{ subspace of finte codimension}\}.$$ Then $\|\cdot\|_\mu$ is a seminorm on $B(X)$ and an algebra seminorm and defines an algebra norm on the Calkin algebra $B(X)/K(X)$.

Both examples are measures of non-compactness, i.e. $\|T\|_\mu = 0$ and $\gamma(T)=0$ iff $T$ is compact.