Infinite dimensional spaces other than functional spaces "Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences.
I don't know if there's "another kind" of infinite dimensional spaces other than space of functions/operators/sequences which is interesting.
 A: In fact all normed spaces are subspaces of some function spaces. This could be the reason why functional analysis has its name.
A: One example might be $M(K)$, the space of all regular Borel measures on $K$ of finite variation, where $K$ is compact space. This space arises as the dual of $C(K)$.
A: $\mathbb{R}$ is an infinite dimensional vector spaces over $\mathbb{Q}$, but it is not an Hilbert space. 
You can see that all the axioms for a vector space are verified if you define the sum of two ''vectors" as the usual sum of real numbers and the product for a scalar $q \in \mathbb{Q}$ as the usual product.
This space has an infinite dimensional Hamel basis. 
And, obviously, any $\mathbb{R}^n$ is finite dimensional over $\mathbb{R}$ but infinite dimensional over $\mathbb{Q}$.
A: An extremely interesting Banach space that admittedly is a sequence space but is not frequently discussed is Tsirelson's space. This is a reflexive Banach space that contains no isomorphic copy of $c_{0}$ or of $\ell_{p}$ for $p\in[1,\infty)$. More commonly, one deals with the dual space $\mathcal{T}$ of Tsirelon's original construction which is, as formulated by Figiel and Johnson, the completion of the set $c_{00}$ of finitely-supported scalar seqeunces with respect to the implicitly defined norm:
\begin{multline*} 
\|x\|_{\mathcal{T}}=\max\left\{\|x\|_{\infty},\frac{1}{2}\sup\left\{\sum_{i=1}^{N}\|E_{i}x\|_{\mathcal{T}}\;\middle\vert\; N\in\mathbb{N} \right.\right. \\ \left.\left. \phantom{\sum_{i=1}^{N}}\text{ and } \{N\}\leq E_{1}<E_{2}<\ldots <E_{N}\right\}\right\} \qquad x=(x_{j})_{j=1}^{\infty}\in c_{00}
\end{multline*}
where $E_{i}\subset\mathbb{N}$ is finite, $E_{i}x=\sum_{j\in E_{i}}x_{j}e_{j}$, and the notation $E<F$ means that $\max(E)<\min(F)$ for subsets $E,F\subset\mathbb{N}$. The Banach space $\mathcal{T}$ is the basis (no pun intended) of numerous counterexamples in Banach space theory and gives rise to a number of variants which have curious properties (such as being arbitrarily distortable).
A: Algebraists study many algebras which are of infinite dimension over their ground field, like Universal Enveloping Algebras, Kac-Moody Algebras, Weyl Algebras, ...
