The set of functions that take in a natural number n and return a REAL number.

    \[ \mathbb{R}^\mathbb{N} = \{f \mid f \colon \mathbb{N} \rightarrow \Re \} \]

Functions of ONE Real Variable (i.e. REAL  to REAL)

    \[ \mathbb{R}^\mathbb{R} = \{f \mid f \colon \Re \to \Re \} \]

Of what use are these vector spaces?

The fundamental theorem (of linear algebra) says that observables (in Quantum Mechanics) are Hermitian Matrices. Which means that the VECTORS form a complete set. That is, the vector space (corresponding to vectors of an observable) is a complete space.

Note that, this is the fundamental theorem of mathematics, not of Quantum Mechanics. Even though it applies perfectly to Quantum Mechanics. What is ever so mysterious about quantum mechanics is that the language of mathematics can be used throughout, without ever stepping into the physical meaning.

For instance, the Uncertainty relation is a basic mathematical property of two non-commuting matrices (observables). It has little to do with Quantum Physics, unless we make the connection to satisfy ourselves.