Introduction to Hilbert Space and the Theory of Spectral Multiplicity: Second Edition

CHAPTER I

THE GEOMETRY OF HILBERT SPACE

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1. Linear Functionals

Throughout this book we shall work with vector spaces over the field of complex numbers, or, as they may be more briefly described, complex vector spaces. The simplest and yet by far the most important example of a complex vector space is the set of all complex numbers, with the vector operations of addition and scalar multiplication interpreted as the ordinary arithmetic operations of addition and multiplication of complex numbers.

We recall an elementary definition. A linear transformation from a complex vector space to a complex vector space is a mapping A from into such that A(αx + βy) = αAx + βAy identically for all complex numbers α and β and all vectors x and y in . Just as the special vector space plays a distinguished role among all complex vector spaces, similarly linear transformations whose range space coincides with (such linear transformations are called linear functionals) play a distinguished role among all linear transformations. Explicitly: a linear functional on a complex vector space is a complex-valued function ξ on such that (and now we proceed, for the sake of variety, to state the definition of linearity in terms slightly different from the ones used above)

(i) ξ is additive (i.e. ξ(x + y) = ξ(x) + ξ(y) for every pair of vectors x and y in ), and

(ii) ξ is homogeneous (i.e. ξ(αx) = αξ(x) for every complex number α and for every vector x in ).

It is sometimes convenient to consider, along with linear functionals, the closely related conjugate linear functionals whose definition differs from the one just given in that the equation ξ(αx) = αξ(x) is replaced by ξ(αx) = α*ξ(x). There is a simple and obvious relation between the two concepts: a necessary and sufficient condition that a complex-valued function ξ on a complex vector space be a linear functional is that ξ* be a conjugate linear functional.

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2. Bilinear Functionals

For the theory that we shall develop, the concept of a bilinear functional is even more important than that of a linear functional. A bilinear functional on a complex vector space is a complex-valued function φ on the Cartesian product of with itself such that if ξy(x) = ηx(y) = φ(x, y), then, for every x and y in , ξy is a linear functional and ηx is a conjugate linear functional.

This definition of a bilinear functional is different from the one commonly used in the theory of vector spaces over an arbitrary field; the usual definition requires that, for every x and y in , both ηx and ξy shall be linear functionals. An example of a bilinear functional in this usual sense may be manufactured by starting with two arbitrary linear functionals ξ and η and writing φ(x, y) = ξ(x)η(y); an obviously related example of a bilinear functional in the sense in which we defined that concept is obtained by writing φ(x, y) = ξ(x)η*(y). The objects that we defined are sometimes called Hermitian bilinear functionals. Further examples of either usual or Hermitian bilinear functionals may be constructed by forming finite linear combinations of examples of the product type described above. After this brief comment on the peculiarity of our terminology (adopted for reasons of simplicity), we shall consistently stick to the definition that was formally given in the preceding paragraph.

It is easy to verify that if φ is a bilinear functional and if the function ψ is defined by ψ(x, y) = φ*(y, x), then ψ is a bilinear functional. A bilinear functional φ is symmetric if φ = ψ, or, explicitly, if φ(x, y) = φ*(y, x) for every pair of vectors x and y. A bilinear functional φ is positive if φ(x, x) 0 for every vector x; we shall say that φ is strictly positive if φ(x, x) > 0 whenever x ≠ 0.

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3. Quadratic Forms

The quadratic form induced by a bilinear functional φ on a complex-vector space is the function defined for each vector x by (x) = φ(x, x). Using this language and notation, we may paraphrase one of the definitions in the last paragraph of the preceding section as follows: φ is positive if and only if is positive in the ordinary sense of taking only positive values.

A routine computation yields the following useful result.

THEOREM 1. If is the quadratic form induced by a bilinear functional φ on a complex vector space , then

for every pair of vectors x and y in .

The process of calculating the values of the bilinear functional φ from the values of the quadratic form , in accordance with the identity in Theorem 1, is known as polarization. As an immediate corollary of this process we obtain (and we state in Theorem 2) the fact that a bilinear functional is uniquely determined by its quadratic form.

THEOREM 2. If two bilinear functionals φ and ψ are such that , then φ = ψ.

Theorem 2 in turn may be applied to yield a simple characterization of symmetric bilinear functionals.

THEOREM 3. A bilinear functional φ is symmetric if and only if is real.

Proof. If φ is symmetric, then (x) = φ(x, x) = φ*(x, x) = *(x) for all x. If, conversely, is real, then the bilinear functional ψ, defined by ψ(x, y) = φ*(y, x), and the bilinear functional φ are such that = ; it follows from Theorem 2 that φ = ψ.

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4. Inner Product and Norm

An inner product in a complex vector space is a strictly positive, symmetric, bilinear functional on . An inner product space is a complex vector space and an inner product in . The vector space of all complex numbers becomes an inner product space if the inner product of α and β is denned to be αβ*; in what follows we shall always interpret the symbol , not merely as a vector space, but as an inner product space with this particular inner product.

It is convenient and, as it turns out, not confusing to use the same notation for inner product in all inner product spaces; the value of the inner product at an ordered pair of vectors x and y will be denoted by (x, y). The quadratic form induced by the inner product also has a universal symbol: its value at a vector x will be denoted by || x ||². The positive square root ||x|| of ||x||² is called the norm of the vector x. Note that the norm of a vector α in the inner product space coincides with the absolute value of the complex number α.

Throughout this book, unless in some special context we explicitly indicate otherwise, the symbol will denote a fixed inner product space; all apparently homeless vectors will be presumed to belong to and the definitions of all concepts and the proofs of all theorems will pertain to .

THEOREM 1. A necessary and sufficient condition that x = 0 is that (x, y) = 0 for all y.

Proof. If (x, y) = 0 for all y, then, in particular, (x, x) = 0 and consequently, since the inner product is strictly positive, x = 0. If, conversely, x = 0, then (x, y) = (0x, y) = 0(x, y) = 0. (Note that the proof of the converse is nothing more than the proof of the fact that if ξ is any linear functional, then ξ(0) = 0. It follows, of course, that if φ is any bilinear functional, then φ(0, y) = φ(x, 0) = 0 for all x and y)

THEOREM 2. (The parallelogram law.) For any vectors x and y,

Proof. Compute.

The reader should realize the relation between Theorem 2 and the assertion that the sum of the squares of the two diagonals of a parallelogram is equal to the sum of the squares of its four sides.

The most important relation between vectors of an inner product space is orthogonality; we shall say that x is orthogonal to y, in symbols x ⊥ y, if (x, y) = 0. In terms of this concept Theorem 1 says that the only vector orthogonal to every vector is 0. For orthogonal vectors the statement of the parallelogram law may be considerably sharpened.

THEOREM 3. (The Pythagorean theorem.) If x ⊥ y, then

The reader should realize the relation between Theorem 3 and the assertion that the square of the hypotenuse of a right triangle is the sum of the squares of its two perpendicular sides.

A family {xj} of vectors is an orthogonal family if xj ⊥ xk whenever j ≠ k. We shall have no qualms about using the obvious inductive generalization of the Pythagorean theorem, i.e. the assertion that if {xj} is a finite orthogonal family, then || Σjxj ||² = Σj || xj ||².

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5. The Inequalities of Bessel and Schwarz

A vector x is normalized, or is a unit vector, if || x || = 1; the process of replacing a non-zero vector x by the unit vector x/|| x || is called normalization. A family {xj} of vectors is an orthonormal family if it is an orthogonal family and each vector xj is normalized, or, more explicitly, if (xj, xk) = δjk for all j and k.

THEOREM 1. (Bessel’s inequality.) If {xj} is a finite orthonormal family of vectors, then

for every vector x.

Proof.

(The expressions (x, xj) will occur frequently in our work; they are called the Fourier coefficients of the vector x with respect to the orthonormal family {xj}.)

It is sometimes useful to realize that the strict positiveness of the inner product is not needed to prove the Bessel inequality. In the presence of strict positiveness, however, the statement of Bessel’s inequality can be improved by adding to it the assertion that equality holds if and only if x is a linear combination of the xj’s. The proof of this addition is an almost immediate consequence of the observation that in the proof of Bessel’s inequality there is only one place at which an inequality sign occurs.

THEOREM 2. (Schwarz’s inequality.) | (x, y) | || x ||·|| y || .

Proof. If y = 0, the result is obvious. If y ≠ 0, write y0 = y/|| y || ; since || y0 || = 1, i.e. since the family consisting of the one term y0 is an orthononormal family, it follows from Bessel’s inequality that | (x,y0) | ||x||

Schwarz’s inequality, just as Bessel’s inequality, would be true even if the inner product were not strictly positive (but merely positive). Our proof of Schwarz’s inequality is not delicate enough to yield this improvement: we made use of strict positiveness through the possibility of normalizing any non-zero vector. In the presence of strict positiveness, however, the statement of Schwarz’s inequality can be improved by adding to it the assertion that equality holds if and only if x and y are linearly dependent; the proof of this addition is, in one direction, trivial and, in the other direction, a consequence of the corresponding facts about Bessel’s inequality.

The Schwarz inequality has an interesting generalization. If {xj} is a non-empty, finite family of vectors, and if γjk = (xj, xk), then the determinant of the matrix [γjk] is non-negative; it vanishes if and only if the xj’s are linearly dependent.

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6. Hilbert Space

THEOREM 1. The norm in an inner product space is

Proof. The strict positiveness of the norm is merely a restatement of the strict positiveness of the inner product. The positive homogeneity of the norm is a consequence of the identity

The subadditivity of the norm follows, using Schwarz’s inequality, from the relations

THEOREM 2. If the distance from a vector x to a vector y is defined to be || x — y || , then, with respect to this distance function, is a metric space.

Proof. The fact that the distance function is strictly positive (i.e. that || x — y || 0, with equality holding if and only if x = y) follows from the strict positiveness of the norm. The fact that the distance function is symmetric (i.e. that || x — y || = ||y — x || for every pair of vectors x and y) follows from the positive homogeneity of the norm and the identity (x — y) = (— 1)(y — x). The validity of the triangle inequality (i.e. the relation || x – y || || x – z || + || z – y || for every triple of vectors x, y, and z) follows from the subadditivity of the norm and the identity x — y = (x — z) + (z — y).

In view of Theorem 2 we shall feel free to use, for inner product spaces, all such topological concepts as convergence, continuity, separability, dense set, closed set, and the closure of a set, and all such metric concepts as uniform continuity, Cauchy sequences,