# Vector Spaces and Linear Transformations

Chapter

## Abstract

Having seen that it was necessary to construct a new theory to describe the behavior of sub-microscopic particles, it should not be surprising that mathematical techniques were introduced concurrently to aid in the development of the new theory.

## Keywords

Vector Space Linear Operator Basis Vector Linear Transformation Product Space
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## References

- 2.See, for example, Paul R. Halmos,
*Finite-Dimensional Vector Spaces*, Van Nostrand, New York, 1958, pp. 3–4.Google Scholar - 17.See, for example,
*Advanced Calculus*, by A. E. Taylor, Ginn and Co., NY, 1955, Chapter 16.Google Scholar - 25.G. F. Roach,
*Green’s Functions, van*Nostrand Reinhold, New York, 1970.Google Scholar - 29.For a good discussion of the different convergence criteria, see F. W. Byron and R. W. Fuller,
*Mathematics of Classical and Quantum Physics*, Vol. I, Addison-Wesley Publishers, Reading, MA, 1969.Google Scholar - 30.A more general mathematical treatment of this problem shows that both 3C and F belong to an
*abstract Hilbert*space. This shows their basic similarity since both the functions and sequences belong to the same abstract space. For a detailed discussion of this point, see W. Schmeidler,*Linear Operators in Hilbert Space*,Academic Press, New York, 1965.Google Scholar - 31.For more details, the reader is referred to treatises such as R. Courant and D. Hilbert,
*Methods of Mathematical Physics*, Vol. I, Interscience Publishers, New York, 1953, pp. 69–81.Google Scholar - 32.For additional discussion see, for example, W. Kaplan,
*Advanced Calculus*, Addison-Wesley, Cambridge, MA, 1952, p. 434.Google Scholar - 33.
*Tables of Integral Transforms*Vols. I-II, edited by A. Erdelyi, McGraw-Hill Book Co., New York, 1954.Google Scholar - 34.This “function” was introduced by P. A. M. Dirac,
*Quantum Mechanics*,Oxford University Press, 1958, pp. 58–62.Google Scholar - E. Meibacher,
*Quantum Mechanics*, 2nd ed., Wiley, New York, 1970, pp. 82–85.Google Scholar - 35.Proper proofs of these properties can be given using Distribution Theory (cf. A. Messiah,
*Quantum Mechanics*, Vol. 1, North-Holland Publishing Co., Amsterdam, 1965, Appendix A). The “proofs” given here are strictly formal proofs.Google Scholar - 38.For additional discussion, see G. Arfken,
*Mathematical Methods for Physicists*, Academic Press, New York, 1968, Sect. 17. 6.Google Scholar - 40.More general mappings of the form a:
*V→V’*are discussed in most advanced books in linear algebra, see, e.g., K. Hoffman and R. Kunze,*Linear Algebra*, Prentice-Hall, Englewood Cliffs, NJ, 1961.Google Scholar - 41.See, for example, J. Indritz,
*Methods in Analysis*, Macmillan, New York, 1963, p. 22.Google Scholar - 44.For a development of the properties and uses of these operators, see, for example, P. O. Löwdin,
*Phys. Rev*.,**97**, 1509 (1955)CrossRefGoogle Scholar - J. V. Neumann,
*Mathematical Foundations of Quantum Mechanics*, Princeton University Press, Princeton, 1955.Google Scholar - 45.

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© Springer-Verlag New York Inc. 1989