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Lecture# Function Spaces and Hilbert Spaces

Description

This lecture covers the concepts of function spaces and Hilbert spaces, starting with the definition of function spaces as vector spaces over C, whose elements are functions. It then introduces the structure of inner product spaces, focusing on the properties of scalar products. The lecture also discusses the importance of completeness in Hilbert spaces, where all Cauchy sequences converge. Examples of inner product spaces, such as the Coordinate space and the set of infinite square summable sequences, are presented. The lecture concludes by highlighting that finite-dimensional inner product spaces are Hilbert spaces, and that most properties of finite-dimensional spaces can be extended to infinite-dimensional Hilbert spaces.

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In course

PHYS-216: Mathematical methods for physicists

This course complements the Analysis and Linear Algebra courses by providing further mathematical background and practice required for 3rd year physics courses, in particular electrodynamics and quant

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In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for arises in this fashion.

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