Functions are Vectors

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Conceptualizing functions as infinite-dimensional vectors lets us apply the tools of linear algebra to a vast landscape of new problems, from image and geometry processing to curve fitting, light transport, and machine learning. Prerequisites: introductory linear algebra, introductory calculus, introductory differential equations. Functions as Vectors Vector Spaces Linear Operators Diagonalization Inner Product Spaces The Spectral Theorem Applications Fourier Series Image Compression Geometry Processing Further Reading Functions as Vectors Vectors are often first introduced as lists of real numbers—i.e. the familiar notation we use for points, directions, and more. $$ \mathbf{v} = \begin{bmatrix}x\\y\\z\end{bmatrix} $$ You may recall that this representation is only one example of an abstract vector space. There are many other types of vectors, such as lists of complex numbers, graph cycles, and even magic squares. However, all of these vector spaces have one thing in common: a finite number of dimensions. That is, each kind of vector can be represented as a collection of $N$ numbers, though the definition of “number” varies. If any $N$-dimensional vector is essentially a length-$N$ list, we could also consider a vector to be a mapping from an index to a value. \[\begin{align*} \mathbf{v}_1 &= x\\ \mathbf{v}_2 &= y\\ \mathbf{v}_3 &= z \end{align*}\ \iff\ \mathbf{v} = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\] What does this perspective hint at as we increase the number of dimensions? Dimensions In higher dimensions, vectors start to look more like functions! Countably Infinite Indices Of course, a finite-length vector only specifies a value at a limited number of indices. Could we instead define a vector that contains infinitely many values? Writing down a vector representing a function on the natural numbers ($\mathbb{N}$)—or any other

Functions are Vectors

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