Functions form a Vector Space

Functions can be treated as infinite dimensional vectors. For example, typically a vector is interpreted as an ordered list of numbers. This could also be interpreted as a function mapping the indices to the values. E.g. a vector in $\mathbb{R}^3$ can be considered to be a function mapping the domain $1, 2, 3$ to the codomain $\mathbb{R}$ (since the values in the vector are real numbers).

If our vector was $N$ dimensional, then this would become: $v: {1, 2, …, N} \to \mathbb{R}$. This could even be defined for infinite $N$ in which case the function becomes a mapping from the natural numbers $\mathbb{N}$ to the real numbers: $v: \mathbb{N} \to \mathbb{R}$.

What if we allowed the “index” to be any real (or complex) number: $v: \mathbb{R} \to \mathbb{R}? This “vector” is just a function from reals to reals. It can be proven¹ that functions, together with the standard addition and scalar multiplication operations form a vector space.

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