Functions Are Vectors (2023)

Exploring how functions can be understood as infinite-dimensional vectors enables the application of linear algebra tools to functional analysis problems. This mathematical insight reveals deep connections between familiar geometric concepts and abstract function spaces, though with important caveats about the limitations of infinite-dimensional spaces.

The Mathematical Joke: From Spherical Cows to Cow-Shaped Functions

The article contains a clever mathematical joke that reverses the traditional “spherical cow” approximation used in physics. Instead of simplifying complex shapes to spheres for easier calculation, the author demonstrates how spherical harmonics—mathematical functions originally developed for spheres—can be generalized to work on actual cow-shaped meshes.

This reversal highlights the power of viewing functions as vectors: techniques developed for simple geometric cases can extend to arbitrarily complex shapes. The cow mesh serves as a humorous example of how mathematical abstractions can return to handle real-world complexity rather than just simplifying it away.

The joke works on multiple levels, celebrating both the practical utility of mathematical generalization and the absurdity of the original “assume a spherical cow” trope that physicists use to make problems tractable.

Vector Spaces: From Finite to Infinite Dimensions

The article begins with familiar finite-dimensional vectors—lists of numbers that can be visualized as arrows in space—then extends this concept to functions. This extension involves two significant conceptual leaps: from natural numbers to real numbers as indices, and from finite to infinite dimensions.

These transitions aren’t trivial mathematical steps. Real numbers represent a “fascinating non-obvious math discovery,” and the passage from finite to infinite dimensions introduces fundamental changes in how mathematical properties behave. What works in finite dimensions doesn’t always hold in infinite-dimensional spaces.

The key insight is that functions satisfy the axioms of vector spaces: they can be added together and multiplied by scalars in ways that preserve the essential algebraic structure. This allows applying linear algebra intuition to functional analysis problems, even when the underlying mathematics becomes more complex.

The Hilbert Space Caveat

The article includes an important disclaimer: “If you’re alarmed by the fact that the set of all real functions does not form a Hilbert space, you’re probably not in the target audience of this post.” This warning highlights a crucial limitation in extending finite-dimensional intuition to function spaces.

Hilbert spaces provide the mathematical framework where inner products, orthogonality, and many other familiar concepts from linear algebra work properly. The set of all real functions is too large and unwieldy to form a proper Hilbert space, meaning some of our geometric intuitions break down.

This limitation doesn’t invalidate the approach—it just requires care in applying finite-dimensional reasoning to infinite-dimensional problems. Many important function spaces (like L² spaces) do form Hilbert spaces and support the full machinery of linear algebra.

Practical Applications: From Theory to Computation

The article demonstrates practical applications through examples like solving differential equations using eigenfunctions and analyzing functions on complex geometric shapes. These applications show how the vector perspective enables computational approaches to problems that would be difficult to handle through traditional analytical methods.

The eigenfunction approach to differential equations exemplifies this power: by viewing solutions as linear combinations of basis functions (eigenvectors), complex partial differential equations become linear algebra problems that computers can solve efficiently.

The extension to arbitrary meshes and shapes shows how this mathematical framework scales from textbook examples to real-world computational problems in graphics, physics, and engineering.

The Conceptual Direction Debate

Discussion reveals disagreement about whether it’s more natural to think of “vectors as functions” or “functions as vectors.” Some argue that vectors are actually discrete-domain functions, making functions the more fundamental concept rather than a generalization of vectors.

This perspective suggests that the familiar finite-dimensional vectors are special cases of the more general function concept, where the domain happens to be a finite set of indices. From this view, the article’s approach reverses the natural conceptual hierarchy.

However, the pedagogical value of starting with familiar vectors and extending to functions remains clear. Most readers have stronger intuition for geometric vectors than for abstract function spaces, making the vector-to-function direction more accessible for building understanding.

Limitations and Breakdown Points

The infinite-dimensional setting introduces fundamental differences from finite-dimensional linear algebra. Properties like compactness behave differently, and operators require careful attention to their domains—concerns that don’t exist in finite dimensions.

The article hints at these limitations when noting that the Laplacian operator is shown to be symmetric rather than self-adjoint, a distinction that matters in infinite dimensions but not finite ones. These technical details reflect deeper mathematical subtleties that can’t be ignored in rigorous applications.

Despite these limitations, the vector perspective provides valuable intuition and computational tools for functional analysis. The key is understanding when the analogy works well and when additional mathematical machinery becomes necessary.

The approach serves as an excellent introduction to functional analysis, where linear algebra intuition helps navigate function spaces while acknowledging the important ways that infinite-dimensional spaces differ from their finite-dimensional counterparts.