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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.
