The direct sum with this norm is again a Banach space.

Compact operators on a Banach space are always completely continuous.

Banach spaces and operator theory, the subject of his thesis.

We may also define a Banach space version of this theorem.

This is a closed subspace of c, and so again a Banach space.

Finally, all of the above holds for integrals with values in a Banach space.

There is a natural analog of this notion in general Banach spaces.

Measures that take values in Banach spaces have been studied extensively.

Several concepts of a derivative may be defined on a Banach space.

Here is the Banach space of sequences converging to zero.

Banach spaces play a central role in functional analysis.

The same name is now used for the analogous construction for a Banach space.

Consider for instance the Banach space l of all bounded real sequences.

The notion of series can be easily extended to the case of a Banach space.

With respect to this norm bv becomes a Banach space as well.

In other areas of analysis, the spaces under study are often Banach spaces.

There are also generalizations to certain types of continuous maps from a Banach space to itself.

This is in stark contrast to the situation in Banach spaces.

One can also study the spectral properties of operators on Banach spaces.

With this norm, the dual space is also a Banach space.

It is necessary that the spaces in question be Banach spaces.

The definition of weak convergence can be extended to Banach spaces.

In particular, the latter is a Banach space.

More generally, this holds in any uniformly convex Banach space.

He was the first to prove the stability of the linear mapping in Banach spaces.