Operations on Hilbert spaces
Given two (or more) Hilbert spaces, we can combine them into a big Hilbert space by taking their direct sum or their tensor product.
Bases
An important concept is that of an orthonormal basis of a Hilbert space H: this is a family {ek}k ∈ B of H satisfying:
- Elements are normalized: Every element of the family has norm 1: ||ek|| = 1 for all k in B
- Elements are orthogonal: Every two different elements of B are orthogonal: <ek, ej> = 0 for all k, j in B with k ≠ j.
- Dense span: The linear span of B is dense in H.
We also use the expressions orthonormal sequence and orthonormal set.
Examples of orthonormal bases include:
- the set {(1,0,0),(0,1,0),(0,0,1)} forms an orthonormal basis of R3
- the sequence {fn : n ∈ Z} with fn(x) = exp(2πinx) forms an orthonormal basis of the complex space L2([0,1])
- the family {eb : b ∈ B} with eb(c) = 1 if b=c and 0 otherwise forms an orthonormal basis of l2(B).
Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter basis is also called a Hamel basis.
Using Zorn's lemma, one can show that every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality. A Hilbert space is separable if and only if it admits a countable orthonormal basis.
Since all infinite-dimensional separable Hilbert spaces are isomorphic, and since almost all Hilbert spaces used in physics are separable, when physicists talk about the Hilbert space they mean any separable one.
If {ek}k ∈ B is an orthonormal basis of H, then every element x of H may be written as
Even if B is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x.
If {ek}k ∈ B is an orthonormal basis of H, then H is isomorphic to l2(B) in the following sense: there exists a bijective linear map Φ : H → l2(B) such that
-
for all x and y in H.
Orthogonal complements and projections
If S is a subset of the Hilbert space H, we define the set of vectors orthogonal to S
Sperp is a closed subspace of H and so forms itself a Hilbert space. If V is a closed subspace of H, then Vperp is called the orthogonal complement of V. In fact, every x in H can then be written uniquely as x = v + w, with v in V and w in Vperp. Therefore, H is the internal Hilbert direct sum of V and Vperp. The linear operator PV : H → H which maps x to v is called the orthogonal projection onto V.
Theorem. The orthogonal projection PV is a self-adjoint linear operator on H of norm ≤ 1 with the property PV2 = PV. Moreover, any self-adjoint linear operator E such that E2 = E is of the form PV, where V is the range of E. For every x in H, PV(x) is the unique element v of V which minimizes the distance ||x - v||.
This provides the geometrical interpretation of PV(x): it is the best approximation to x by elements of V.
Reflexivity
An important property of any Hilbert space is its reflexivity. In fact, more is true: one has a complete and convenient description of its dual space (the space of all continuous linear functions from the space H into the base field), which is itself a Hilbert space. Indeed, the Riesz representation theorem states that to every element φ of the dual H' there exists one and only one u in H such that
-
for all x in H and the association φ ↔ u provides an antilinear isomorphism between H and H'. This correspondence is exploited by the bra-ket notation popular in physics but frowned upon by mathematicians.
Bounded operators
For a Hilbert space H, the continuous linear operators A : H → H are of particular interest. Such a continuous operator is bounded in the sense that it maps bounded sets to bounded sets. This allows to define its norm as
The sum and the composition of two continuous linear operators is again continuous and linear. For y in H, the map that sends x to <y, Ax> is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form
This defines another continuous linear operator A* : H → H, the adjoint of A.
The set L(H) of all continuous linear operators on H, together with the addition and composition operations, the norm and the adjoint operation, forms a C*-algebra; in fact, this is the motivating prototype and most important example of a C*-algebra.
An element A of L(H) is called self-adjoint or Hermitian if A* = A. These operators share many features of the real numbers and are sometimes seen as generalizations of them.
An element U of L(H) is called unitary if U is invertible and its inverse is given by U*. This can also be expressed by requiring that <Ux, Uy> = <x, y> for all x and y in H. The unitary operators form a group under composition, which can be viewed as the automorphism group of H.
Unbounded operators
In quantum mechanics, one also considers linear operators which need not be continuous and which need not be defined on the whole space H. One requires only that they are defined on a dense subspace of H. It is possible to define self-adjoint unbounded operators, and these play the role of the observables in the mathematical formulation of quantum mechanics.
Typical examples of self-adjoint unbounded operators on the Hilbert space L2(R) are given by the derivative Af = if ' (where i is the imaginary unit and f is a square integrable function) and by multiplication with x: Bf(x) = xf(x). These correspond to the momentum and position observables, respectively. Note that neither A nor B is defined on all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of L2(R).
See also
References
- Paul Halmos, Measure Theory, D. van Nostrand Co, 1950.
- Jean Dieudonné, Foundations of Modern Analysis, Academic Press, 1960.
- Hermann Weyl, The Theory of Groups and Quantum Mechanics, Dover Press, 1950. This book was originally published in German in 1931.
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