Jeremy J. Becnel was born in the rural community of Kraemer, Louisiana, in 1979 and completed his undergraduate studies in mathematics and computer science at Nicholls State University in May 2001. He earned a Master of Science degree from Louisiana State University in December 2002 and a PhD in August 2006. Immediately after graduation, Dr. Becnel began working at Stephen F. Austin State University (SFASU). He has worked at SFASU for fifteen years as a professor in the Department of Mathematics and the Department of Computer Science. During this time, Dr. Becnel has supervised numerous undergraduate research projects and graduate theses. He is a recipient of the National Science Foundation Division of Undergraduate Research grant and a Young Investigator's Grant from the National Security Agency, among others. Through the former grant, he has worked with colleagues to create CalcVR, a virtual reality app for multivariable calculus. His research interests include quantum computing, Radon transforms, probability theory, white noise analysis, and topological vector spaces. Dr. Becnel currently lives in the small town of Nacogdoches, Texas with his wife, Kesha, along with their daughters, Alyssa and Anna.

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• 1. Schwartz Space. 1.1. The Schwartz Space. 1.2. The Schwartz Norm Topology. 1.3. Fourier Transform. 1.4. Physical Context. 1.5. Orthonormal Basis, Creation, and Annihilation Operators in L²(R). 1.6. Number Operator and the Spaces Sp(R). 1.7. The Projective Limit Topology on S(R). 1.8. L² norms on S(R). 1.9. Equivalence of the Topologies. 1.10. The Sequence Space. 1.11. Multidimensional. 2. Nuclear Spaces. 2.1. Topological Vector Spaces. 2.2. Countably-Normed Spaces. 2.3. Nuclear Space Structure. 2.4. Nuclear Space Construction from a Hilbert Space and an Operator. 3. Dual Space. 3.1. Linear Functionals. 3.2. Dual Space. 3.3. The Dual of a Nuclear Space. 3.4. Topologies on Hʼ. 3.5. Reflexivity. 3.6. Convergence and Completeness in Hʼ. 3.7. Properties of the Topology on H' and H_p. 3.8. Inductive Limit Topology. 3.9. Sequential Space. 3.10. First Countability (Lack Thereof). 3.11. Summary. 3.12. Abstract Kernel Theorem. 3.13. White Noise Dual. 3.14. Schwartz Space Dual. 3.15. Boxes and Continuous Functions. 4. Probability Theory. 4.1. Probability Theory Fundamentals. 4.2. Product Sigma Algebras in Infinite Dimensions. 4.3. Sigma Algebras on the Dual of a Nuclear Space. 4.4. Probability Measures on Infinite Dimensional Spaces. Appendix: Banach and Hilbert Spaces.

Jeremy J. Becnel was born in the rural community of Kraemer, Louisiana, in 1979 and completed his undergraduate studies in mathematics and computer science at Nicholls State University in May 2001. He earned a Master of Science degree from Louisiana State University in December 2002 and a PhD in August 2006. Immediately after graduation, Dr. Becnel began working at Stephen F. Austin State University (SFASU). He has worked at SFASU for fifteen years as a professor in the Department of Mathematics and the Department of Computer Science. During this time, Dr. Becnel has supervised numerous undergraduate research projects and graduate theses. He is a recipient of the National Science Foundation Division of Undergraduate Research grant and a Young Investigator's Grant from the National Security Agency, among others. Through the former grant, he has worked with colleagues to create CalcVR, a virtual reality app for multivariable calculus. His research interests include quantum computing, Radon transforms, probability theory, white noise analysis, and topological vector spaces. Dr. Becnel currently lives in the small town of Nacogdoches, Texas with his wife, Kesha, along with their daughters, Alyssa and Anna.

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• 1. Schwartz Space. 1.1. The Schwartz Space. 1.2. The Schwartz Norm Topology. 1.3. Fourier Transform. 1.4. Physical Context. 1.5. Orthonormal Basis, Creation, and Annihilation Operators in L²(R). 1.6. Number Operator and the Spaces Sp(R). 1.7. The Projective Limit Topology on S(R). 1.8. L² norms on S(R). 1.9. Equivalence of the Topologies. 1.10. The Sequence Space. 1.11. Multidimensional. 2. Nuclear Spaces. 2.1. Topological Vector Spaces. 2.2. Countably-Normed Spaces. 2.3. Nuclear Space Structure. 2.4. Nuclear Space Construction from a Hilbert Space and an Operator. 3. Dual Space. 3.1. Linear Functionals. 3.2. Dual Space. 3.3. The Dual of a Nuclear Space. 3.4. Topologies on Hʼ. 3.5. Reflexivity. 3.6. Convergence and Completeness in Hʼ. 3.7. Properties of the Topology on H' and H_p. 3.8. Inductive Limit Topology. 3.9. Sequential Space. 3.10. First Countability (Lack Thereof). 3.11. Summary. 3.12. Abstract Kernel Theorem. 3.13. White Noise Dual. 3.14. Schwartz Space Dual. 3.15. Boxes and Continuous Functions. 4. Probability Theory. 4.1. Probability Theory Fundamentals. 4.2. Product Sigma Algebras in Infinite Dimensions. 4.3. Sigma Algebras on the Dual of a Nuclear Space. 4.4. Probability Measures on Infinite Dimensional Spaces. Appendix: Banach and Hilbert Spaces.