Hilbert spaces are generalising some concepts of two- or three-dimensional Euclidean space to larger vector spaces of possibly infinite dimensions. Most notably taking limits or computing lengths and angles is possible in the same way as for Euclidean geometry, thus allowing to perform vector calculus or to numerically approximate in a sound way. Same as vector spaces, Hilbert spaces are defined with respect to a field F, see definition below. In our case F can be typically identified with the field of all complex numbers C or the real numbers R.
The first ingredients to a Hilbert space are ways to measure angles and distances, i.e. an inner product and a norm.
Definition 2.1. An inner product space over a field F is a vector space V (over the
same field) that is further equipped with an inner product, i.e. a map h · | · iV : V × V → F
that satisfies(for all vectors x, y, z ∈ V and all α ∈ F)
hx|yi∗V = hy|xiV (conjugate symmetry) (2.18)
hx|αy + ziV = α hx|yiV + hx|yiV (linearity in the last argument) (2.19)
hx|xiV ≥ 0 and hx|xiV = 0 ⇒ x = 0 (positive-definiteness), (2.20)
where the asterisk “∗” denotes complex conjugation. We typically drop the “V ” subscript
2.2. ELEMENTS OF FUNCTIONAL ANALYSIS 13
Remark 2.2. Some literature uses a deviating definition for the inner product, where
not the second, but the first argument in (2.19) is linear, i.e. where (2.19) would be replaced by
hαy + z|xiV = α hy|xiV + hz|xiV .
Our definition is in better agreement with the usual convention of quantum physics and quantum chemistry due to the resemblance of Dirac notation [49].
Definition 2.3. Given a vector space V over the field F, a norm is a map k · k : V → R
such that the following axioms hold for all vectors x, y ∈ V and all α ∈ F:
kαxk = |α| kxk (absolute scalability) (2.21)
kx + yk ≤ kxk + kyk (triangle inequality) (2.22)
If kxk = 0 ⇒ x is the zero vector (norm separates points) (2.23)
If such a norm can be found for a particular vector space V , one typically refers to V as a normed vector space as well.
Proposition 2.4. For every inner product space exists the so-called induced norm
kxkV =
q
hx|xiV ∀x ∈ V. (2.24)
One may drop the subscript on the norm if it is clear from context. Proof. See [49].
The second ingredient for a Hilbert space is a property called completeness. Form- ally it is defined as such:
Definition 2.5. A vector space V is called complete if every Cauchy sequence of vectors in V has a limit in V .
Let us first recall, that a sequence (xn)n∈N∈ V is called Cauchy if
∀ε > 0 ∃M ∈ N such that kxn− xmkV < ε ∀n, m > M.
One can show that every converging sequence is Cauchy. A roughly equivalent way of
phrasing definition 2.5 is therefore, that a space V is complete iff every sequence (xn) of
elements which come arbitrarily close at large enough n tend towards an element, which is from V as well.
Example 2.6. To make the concept of completeness more clear, let us consider a
counterexample. For this let us leave the setting of vector spaces and more broadly think
about sequences defined on sets of numbers4, where the concept of completeness applies
as well.
It is well known that the sequence
xn= n X k=0 1 k! ∈ Q 4
This is fine, since completeness is in fact a property on so-called metric spaces, which are related to normed vector spaces, but have much less structure.
14 CHAPTER 2. MATHEMATICAL FOUNDATION OF QUANTUM MECHANICS converges to Euler’s number e, i.e.
lim
n→∞xn= e 6∈ Q.
In other words Q is not complete.
One may, however, build the completion of Q by just including all limiting points of all sequences in Q. In fact this is one way of defining the set of real numbers R.
Remark 2.7. A subtle point about completeness is that it depends on the norm which
is used to determine whether a sequence is Cauchy or not. In other words a vector space may be complete with respect to one norm, but not with respect to another. Similarly the completion of a space with respect to two different norms may yield different spaces.
In practice the choice of the norm is only of importance for infinite-dimensional vector spaces, since for finite-dimensional real or complex vector spaces all norms are
equivalent5anyway.
Finally we can state
Definition 2.8. A Hilbert space H is an inner product space, which is complete with
respect to the induced norm.
In other words a Hilbert space is a space, where the inner product naturally defines a way to measure distances and take limits, that is perform calculus. Thinking ahead towards the integral and differential operators we will define on such Hilbert spaces, this is exactly what we will need.
Before we look into some Hilbert spaces relevant for QM, let us first clarify the concept of denseness and separability.
Definition 2.9. A subspace S of a vector space V is called dense in V if each vector
x∈ V either is a member of S or one may find a Cauchy sequence in S for which x is
the limit point.
In other words S is dense in V if we can — for each element of V — construct a sequence of approximations inside the smaller space S, representing the desired element up to arbitrary accuracy. Denseness is therefore one of the fundamental properties required for approximation.
Example 2.10. Returning to example 2.6 on the previous page we note, that Q is dense in R. This guarantees that we may approximate any real number up to arbitrary accuracy by an appropriate sum of fractions, which is one of the assumptions behind any floating point operation performed on the computer.
Definition 2.11. A Hilbert space is separable6 iff it admits a countable orthonormal
basis.
Remark 2.12. If a Hilbert space is separable we can find a basis set7 {ϕµ}µ∈Ibas of at
most countably infinite cardinality, i.e. where Ibas⊆ N. With this we can write for each
Ψ ∈ H:
Ψ = X
µ∈Ibas
cµϕµ. (2.25)
5That is they induce the same topology.
6In the broader context of metric spaces, a separable space has a countable, dense subset. 7
This remark sketches the construction of a so-called Schauder basis, which is related, but not identical to the concept of a Hamel basis, which is usually employed in finite-dimensional linear algebra.
2.2. ELEMENTS OF FUNCTIONAL ANALYSIS 15
This in turn uniquely identifies each Ψ with a sequence (cµ)µ∈Nof complex numbers. By
this means each complex, separable Hilbert space is isomorphic to the space of complex-
valued, square-summable sequences l2(N, C). One can easily show that this isomorphism
is even an isometry, i.e.
kΨkH = k(cµ)kl2 = v u u t ∞ X µ=0 |cµ|2.
By transitivity all separable Hilbert spaces are isometrically isomorphic.
In our remaining discussion we will only encounter complex, separable Hilbert spaces. This implies:
• If Ψ ∈ H is a vector in a Hilbert space, we can always identify it with a (possibly infinite) column vector of complex coefficients.
• Finite-dimensional Hilbert spaces are isomorphic to Cd, where d is the dimension-
ality. Their vectors are thus identified by a column of complex numbers of finite size.
Remark 2.13. A consequence of remark 2.12 is that we can numerically approximate
all separable Hilbert spaces rather naturally. For example by restricting the sum in (2.25) to only a finite number of d basis functions, we can make sure that the resulting Ψ
is located in only a d-dimensional subspace H(d)
⊂ H. Moreover this subspace is dense,
since in the limit of taking all basis functions, we get exactly H. In turn since H(d)is
finite-dimensional, we can identify each approximation to Ψ with a vector in Cd, which
can be represented numerically on the computer, regardless of the structure of H.