4.2.1 Basis for Banach space
A sequence of vectors {x1, x2, x3, . . .} in an infinite dimensional Banach space X is said to be
a Schauder basis for X if to each vector x in the space there corresponds a unique sequence of scalars {c1, c2, c3, . . .} such that [87, p.1]
x =
∞
X
n=1
cnxn (4.2.1)
The convergence of the series is understood to be with respect to the strong norm topology of X, i.e. for all x, °
° ° ° °x − n X i=1 cixi ° ° ° ° °→ 0 as n → ∞ (4.2.2)
For an infinite-dimensional Banach space, the term basis shall refer to a Schauder basis; a Banach space with a basis must be separable. While almost all infinite dimensional separable Banach spaces are known to have bases, this is not always the case, as demonstrated by Per Enflo [88]. For practical purposes, and certainly this text, separability is assumed. The coefficients are linear functionals of the element x ∈ X such that ci = Ψi(x), where Ψi(x)
describes the required function.
1A norm kf k is a semi-norm |f | with the additional property that it is zero if and only if f is the zero
4.2.2 Basis for Inner-Product and Hilbert Space
Banach Space does not include the structure provided by an inner product, i.e. for the case of an inner-product space. It has a norm, and hence, the topology required to define ”closeness” of elements. Intuitively, inner-product is a measure of orthogonality between two points in the space. Various forms can be used and are typically a linear combination of the products of elements (points in the space). Though not required, inner-products and norms are often simply related, for instance, kf, gk2 = hf, gi (the norm is induced by the inner-product). The definition of the inner-product is chosen for convenience, but is usually stated as a linear combination of weighted element products, such as
hf, gi =X
i
wifigi or
Z
wifi(x)gi(x)dx
where {wi} denotes a set of weighting coefficients, appropriately selected [89, pp. 307-318].
Inner-product spaces are also known as Pre-Hilbert spaces (inner-product spaces that are not complete). An inner product space that is complete in the norm kxk = hx, xi1/2 is a Hilbert
Space (c.f. section 4.1.1). Orthogonality of two elements f and g requires hf, gi = 0 for f 6= g. If in addition, f = g, orthonormality requires kf k = 1 and hf, gi = 1. In a distributional sense, hxi, xji = δij for xi, xj ∈ X.
For an orthonormal set of points, {xi} in an inner product space X, the set {xi} is linearly
independent. There is a maximal orthonormal set B in X with xi ⊂ B [89, p. 306]. Of
course, X generalizes to Hilbert Spaces.
It follows that an orthonormal set B = {xi} in an inner product space X is maximal if and
only if x⊥xi for all i implies that x = 0. A maximal orthonormal set B in a Hilbert Space
H is referred to as an orthonormal basis for H. We also note that a Hilbert Space H has a
countable orthonormal basis if and only if it is separable [89, p. 314].
A basis for a Hilbert space is a Riesz basis if it is equivalent to an orthonormal basis, that is, if it is obtained from an orthonormal basis by means of a bounded invertible operator [87, p.26]. In this section, we have addressed Q5 on the Eigenmode Expansion Method, from chapter 2, namely: How do Riesz bases relate to the concepts we have already discussed, such as orthonormal bases, or Schauder bases?
4.2.3 Eigensystems and Root Systems
For a linear operator T , with domain D(T ) and range R(T ) contained in linear space X. A scalar λ ∈ C such that there exists an x ∈ D(T ), x 6= 0 satisfying the equation T x = λx is an
The null space of the transformation (T − λI), N (T − λI) is the eigenmanifold or eigenspace corresponding to eigenvalue λ. Note that this definition applies to linear operators, whether continuous or not.
The complex spectral theorems for finite dimensional Hilbert spaces are well-understood; for
T : Hn→ Hn, n-dimensional space Hn has an orthonormal basis of eigenvectors {x
n} if and
only if T is normal (c.f section 4.4.2). If T : Hn → Hn admits a diagonal representation then any nondiagonal representation (i.e. in terms of a basis that is not an eigenbasis) can be diagonalized. That is,
[T ]e= diag[λ1, λ2· · · λn] for eigenbasis e = {xn}
The Spectral Theorem, for the finite-dimensional case is thus a generalization of the familiar theorem from linear algebra that a self-adjoint matrix can be diagonalized. Furthermore, that there is a diagonal matrix D and a unitary matrix U such that T = U DU−1. Diago-
nal components of D are eigenvalues of T listed in some order, repeated according to their multiplicity [90, p. 52].
Extended to infinite-dimensional spaces, where T : H → H is a compact, self-adjoint (or normal) linear operator acting on Hilbert Space H. There exists an orthonormal system of eigenvectors {un} corresponding to nonzero eigenvalues {λn} such that every x ∈ H can be
uniquely represented as x = x0+ ∞ X n=1 hx, uniun (4.2.3)
where x0 satisfies T x0 = 0. Furthermore, if {λn} is an infinite set of distinct eigenvalues,
then limn→∞λn = 0. The Hilbert-Schmidt theorem for compact self-adjoint operators [91,
pp. 179-180] defines expansion
T x =X
n
λnhx, xnixn (4.2.4)
An isolated eigenvalue λ is called a normal eigenvalue if its algebraic multiplicity is finite and the Hilbert space can be decomposed into the direct sum of subspaces H = Lλu Rλ. Lλ is
the root subspace of A and Rλ is an invariant subspace for A in which (T − λI)−1 exists. The
root space Lλ is the space of all eigen and root vectors of T corresponding to λ [92, p.10].
If the elements {uk} correlate with
T up = λ0up+ up−1, p = 1, 2, . . . , k (4.2.5)
then element uk is called a k-associated vector to the eigenvector u0. The number k + 1 is the length of the chain u1, u2, . . . , uk. The element u0 is called an eigenvector of rank r if the
Figure 4.1: Surface S enclosing volume V2, embedded in volume V1
chain, and the elements are linearly independent [92, p.1-10]. Eigenvectors and associated
vectors are joined under the common name of root vectors.
In the case of the nonselfadjoint, compact linear operator T , it is not clear if the operator has root vectors. Furthermore, it is not clear whether either the eigenvectors, the root vectors, both or neither are complete and form a basis in H. In the case where operator T is normal, the spectral theorem in equation (4.2.4) applies. We consider the nonnormal case in section 4.4.2.