Linear operators also play a great role in the infinite-dimensional case.

The above discussion can be extended to the case of when the state space is infinite-dimensional with virtually nothing changed.

However, such a curve may not always exist in infinite-dimensional spaces.

The solution, however, is infinite-dimensional in the general case.

That is, we are not interested in estimating the infinite-dimensional component.

It is however much more general as there are important infinite-dimensional Hilbert spaces.

It follows that the group of symplectomorphisms is always very large, and in particular, infinite-dimensional.

It does however have a well-known family of infinite-dimensional unitary representations.

Any given system is identified with some finite- or infinite-dimensional Hilbert space.

The Lie algebra is denoted by and may be infinite-dimensional.

Such systems are therefore also known as infinite-dimensional systems.

The most important distinction is between finite-dimensional representations and infinite-dimensional ones.

The basic idea is to replace the infinite-dimensional linear problem:

Note however that weak and strong topologies are always distinct in infinite-dimensional space.

The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context, see below.

In contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

The groups involved will be infinite-dimensional in almost all cases - and not Lie groups - but the philosophy is the same.

Infinite-dimensional optimization problems can be more challenging than finite-dimensional ones.

In mathematics, infinite-dimensional holomorphy is a branch of functional analysis.

Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions.

The collection of possible candidates for is infinite-dimensional.

On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent.

On the other hand, infinite-dimensional Montel spaces are never normable.

The set of all wavefunctions for a given system is an infinite-dimensional complex Hilbert space.

It can be shown that there is no infinite-dimensional analogue of Lebesgue measure.