Table of Contents Frontmatter Chapter 1. Notation and Preliminaries Abstract. This preliminary chapter is a quick-reference guide to the notation, terminology, and background information that will be assumed throughout the volume. Unlike the rest of the text, in this preliminary chapter we may state results without proof or without the motivation and discussion that is provided throughout the later chapters.
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Much of what we do in real analysis centers on issues of convergence or approximation. What does it mean for one object to be close to or to approximate another object? How can we define the limit of a sequence of objects that appear to be converging in some sense? We studied metric spaces in detail in Chapter 2.
A metric on a set X provides us with a notion of the distance between points in X.
In this chapter we will study norms, which are special types of metrics. However, in order for us to be able to to define a norm, our set X must be a vector space.
In this chapter we will take a closer look at normed and Banach spaces. Moreover, because we also have a notion of limits we can go even further and define infinite series of vectors which are limits of partial sums of the series. Our focus in much of Chapters 2—5 was on particular vector spaces and on particular vectors in those spaces.
Most of those spaces were metric, normed, or inner product spaces.
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Now we will concentrate on classes of operators that transform vectors in one space into vectors in another space. In this chapter we will study operators that map one Hilbert space into another. In contrast all finite-dimensional inner product spaces over R or C , such as those used in quantum computation , are automatically metrically complete and hence Hilbert spaces. In some cases we need to consider non-negative semi-definite sesquilinear forms.
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We show how to treat these below. A simple example is the real numbers with the standard multiplication as the inner product. More generally, the real n -space R n with the dot product is an inner product space, an example of a Euclidean n -space. The general form of an inner product on C n is known as the Hermitian form and is given by.
For the real case this corresponds to the dot product of the results of directionally different scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. Up to an orthogonal transformation it is a weighted-sum version of the dot product, with positive weights.
Operator Theory in Inner Product Spaces | SpringerLink
The article on Hilbert space has several examples of inner product spaces wherein the metric induced by the inner product yields a complete metric space. An example of an inner product which induces an incomplete metric occurs with the space C [ a , b ] of continuous complex valued functions f and g on the interval [ a , b ]. The inner product is. This sequence is a Cauchy sequence for the norm induced by the preceding inner product, which does not converge to a continuous function. For real random variables X and Y , the expected value of their product. This definition of expectation as inner product can be extended to random vectors as well.
However, inner product spaces have a naturally defined norm based upon the inner product of the space itself that does satisfy the parallelogram equality:. This is well defined by the nonnegativity axiom of the definition of inner product space. The norm is thought of as the length of the vector x. Directly from the axioms, we can prove the following:.
Let V be a finite dimensional inner product space of dimension n. Recall that every basis of V consists of exactly n linearly independent vectors. Using the Gram—Schmidt process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm.
This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way.
Let V be any inner product space. Then a collection. We say that E is an orthonormal basis for V if it is a basis and. Any separable inner product space V has an orthonormal basis. Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that. Any complete inner product space V has an orthonormal basis. The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below.
This completes the proof. Parseval's identity leads immediately to the following theorem:. Then the map. This theorem can be regarded as an abstract form of Fourier series , in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials. Note that the underlying index set can be taken to be any countable set and in fact any set whatsoever, provided l 2 is defined appropriately, as is explained in the article Hilbert space. In particular, we obtain the following result in the theory of Fourier series:. Then the sequence indexed on set of all integers of continuous functions.
The mapping. Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1.
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This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials. Several types of linear maps A from an inner product space V to an inner product space W are of relevance:. From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces.
A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces. Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.
This construction is used in numerous contexts.
Operator Theory in Inner Product Spaces
The Gelfand—Naimark—Segal construction is a particularly important example of the use of this technique. Another example is the representation of semi-definite kernels on arbitrary sets. This generalization is important in differential geometry : a manifold whose tangent spaces have an inner product is a Riemannian manifold , while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo-Riemannian manifold. By Sylvester's law of inertia , just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with nonzero weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index.
Product of vectors in Minkowski space is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. The term "inner product" is opposed to outer product , which is a slightly more general opposite. Note that the outer product is defined for different dimensions, while the inner product requires the same dimension.