Attempting to define a well-behaved "infinite-dimensional determinant" for all operators will get us into trouble fairly quickly. All the exterior powers are also infinite-dimensional. There are modifications of the notion of Fredholm determinant for operators on Hilbert space which differ from the identity by an operator from a von Neumann-Schatten ideal. A related notion is the one of a von Koch determinant defined for some classes of infinite matrices.
For all this see.
Linear Mathematics in Infinite Dimensions
Gohberg, I. Introduction to the theory of linear nonselfadjoint operators. Translations of Mathematical Monographs. The answers above point out that one cannot define a determinant in a meaningful way on the algebra of bounded operators on a Banach space, unless finite-dimensional. This is one of those articles which changed the face of mathematics forever.
Springer GTM, Home Questions Tags Users Unanswered. Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions? Ask Question. Xuxu Xuxu 2 2 silver badges 8 8 bronze badges. We will denote the norm on by. Notice that if one applies this definition to , the norm of a point is just its distance in the origin.
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So we think of a norm as a function that assigns lengths to vectors in our vector space. In general, a norm is any function that satisfies the following three axioms:. One can verify that any inner product induces a norm. Although we defined the norm in terms of an inner product, we say that any function satisfying 1 , 2 , and 3 is a norm, whether or not it is given in terms of an inner product. So, any inner product defines a norm, but not every norm is given by an inner product.
Infinite Dimensional Linear Control Systems, Volume 201
For example, it is impossible to define an inner product on such that the induced norm is. We need one more definition before we can define a Hilbert Space. We need the concept of completeness. This is a fundamental property of the real numbers — completeness is what allows us to do real analysis. Such a gap does not exist in , so we say the reals are complete. We are now in a position to define a Hilbert Space: a Hilbert Space is a complete vector space equipped with an inner product.
A similar structure is a Banach Space, which is a complete vector space equipped with a norm. So any Hilbert Space is a Banach Space, but the converse is not true. We can immediately get our hands on some Hilbert Spaces: and are both finite-dimensional Hilbert Spaces. These are not particularly interesting Hilbert Spaces because they are finite-dimensional. As we stated before, is a Hilbert Space.
We can also readily see that has no finite basis. Indeed, an example of a basis for is the collection of sequences where the appears in the entry. The upshot is that we can work exclusively in without sacrificing the generality obtained by referring to a general Hilbert Space. Since is a vector space, the natural thing to do is think about linear transformations of the space.
We define a linear operator on in the same way a linear transformation is defined in linear algebra.
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It should be noted that not everything one may have learned about linear transformations in linear algebra is true for linear operators on. For example, consider the shift operators and on defined by and. It is easily verified that these are both linear operators, and that is injective but not surjective, is surjective but not injective, and but.
In linear algebra, one learns that all of these conditions are equivalent, but in Hilbert Space this is not the case. An important part of operator theory is determining what kinds of operators on behave like linear transformations on a finite-dimensional vector space. We call a linear operator on bounded if there is a constant such that is bounded on the unit ball by. The norm of a linear operator is defined to be the smallest such that works in the preceding definition.
Equivalently, is the largest value of , where ranges over the unit ball in. An interesting fact about linear operators on is that they are continuous if and only if they are bounded an exercise!
Linear Mathematics in Infinite Dimensions | Department of Mathematics
We define to be the set of all bounded continuous linear operators on. Our goal is to define and understand some topologies on. This should be a bit surprising. There are many topologies that can be put this set, but we will consider the three most common ones: the norm topology, the strong operator topology, and the weak operator topology. Instead we can define a topology by describing what properties our set has when equipped with this topology. We can also define a topology in terms of a base which, roughly speaking, is a collection of open sets that generates the rest of the open sets by taking unions.
It may not seem obvious that changing the topology will have a big effect on convergence of sequences; after all, the definition of convergence that one meets in a real analysis course does not explicitly mention open sets! In real analysis, convergence is usually defined in regards to a metric. In more general topological spaces, there may be no metric although in Hilbert Space the metric is induced by the norm , so the definition from real analysis may not be applicable. Take an open interval about 1 on the vertical axis and eventually, all but finitely many points are in this interval.
Linear Mathematics in Infinite Dimensions
Let be a sequence in a topological space. We say if for every open set containing , we have for sufficiently large. Microsoft and. Mobile Computing. Networking and Communications.
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