We started by reviewing the concepts of Vectors and Inner Products in 3 Dimensions. These included the concepts of norm, unit vectors, linear combinations, inner product, orthogonality, and an orthonormal set. It was shown how every vector can be written as a linear combination of the orthonormal set, and how to obtain the projections of a vector onto the orthonormal set.
Presented in this way, it was easy to extend these concepts into N Dimensions.
Next, by considering a simple step function, it was easy to extend these vector concepts to apply to functions considered as vectors in 3-Dimensions.
Finally, a giant conceptual leap was made which allows us to look at continuous functions, defined on a finite interval, to be functions in a "continuous dimensional space", where the number of dimensions is uncountably infinite. The inner product becomes an integral, by which orthonormal sets of functions can be defined.
The decomposition of a function into its components along an orthonormal set of functions is called a "Generalized Fourier Series". Given that the number of dimensions of the vector space is uncountably infinite (like the Real numbers), but the number of basis functions in the orthonormal set is countably infinite (like the Integers), it is always possible that a given function may not be representable in terms of the basis. We have to be careful of this and remember that the Fourier series is not necessarily a strict equality.
Some exercises were given, which will be part of assignment #02, due next Wednesday.