This discussion focuses on introducing inner product spaces with real-world examples, emphasizing their significance in quantum mechanics and Hilbert spaces. The dot product in Euclidean spaces serves as a tangible introduction, while extending this concept to matrices, polynomials, and functions is explored. The Schrödinger equation in quantum mechanics exemplifies an eigenvalue problem where eigenvectors reside in a Hilbert space. The Gram-Schmidt process is highlighted as a crucial method for obtaining an orthonormal basis using inner products.
PREREQUISITESMathematicians, physicists, and students seeking to understand the applications of inner product spaces in real-world scenarios, particularly in quantum mechanics and linear algebra.
matqkks said:What is the most motivating way to introduce general inner product spaces? I am looking for examples which have a real impact. For Euclidean spaces we relate the dot product to the angle between the vectors which most people find tangible. How can we extend this idea to the inner product of general vectors spaces such as the set of matrices, polynomials, functions?