Functions are vectors

Conceptualizing functions as infinite-dimensional vectors allows us to apply the tools of linear algebra to various new problems, such as image and geometry processing, curve fitting, light transport, and machine learning. This approach requires a basic understanding of linear algebra, calculus, and differential equations. Vectors can be thought of as lists of numbers, but they can also represent mappings from an index to a value. In higher dimensions, vectors start to resemble functions. We can represent functions on countably infinite domains by extending the list indefinitely, but for functions defined on uncountably infinite domains, we need to view vectors as arbitrary functions. Functional analysis formalizes the representation of functions as infinite-dimensional vectors. In this post, we will focus on building intuition through analogies to finite-dimensional linear algebra. A vector space is defined by a set of vectors, a scalar field, and certain operations for addition and scalar multiplication. We can define a vector space of real functions by considering the set of functions from the real numbers to the real numbers. Function addition and scalar multiplication follow natural rules of function application and scaling. We can prove that this set of functions forms a vector space by satisfying all the vector space axioms. The standard basis for functions consists of basis functions that have

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