Does there exist a real Hilbert space with countably infinite dimension as a vector space over $\mathbb{R}$?

Essentially what the title says – where to me a Hilbert space is a complete (Hermitian) inner product space, am I safe to assume every such real Hilbert space is of uncountable dimension over $\mathbb{R}$, or is there a countable-dimension example?

Thanks a lot 🙂

Solutions Collecting From Web of "Does there exist a real Hilbert space with countably infinite dimension as a vector space over $\mathbb{R}$?"

An infinite dimensional (real) Hilbert space has dimension at least $\mathfrak{c}=2^{\aleph_0}=|\mathbb{R}|$ as a vector space. One way to see this is by taking an orthornormal sequence $e_1,e_2,\ldots$, and considering the linearly independent set $\{\sum_{k=1}^\infty t^ke_k:0\lt t\lt 1\}$.

The same fact extends to Banach spaces, but there orthogonality cannot be used to write so short of a proof. A proof is given in this short article by Lacey.

To just see that the dimension cannot be countable, you could use Baire’s theorem.

For more on this in the Hilbert space case, see Problem 7 of Halmos’s Hilbert space problem book. (It is assumed there that the Hilbert spaces are complex instead of real, but this does not affect your question.)

I think natural example of Hilbert space with a countable basis is a space of $L^2([-\pi,\pi])$ with a natural fourier set of basis functions ${e^{inx}, i \in -\infty, \ldots, \infty}$. See https://en.wikipedia.org/wiki/Fourier_series