This is a followup to the answers here and here. I have not seen this term in any textbooks I have or many online resources. It is not, for example, present on the SVM wikipedia page.
What is a hypothesis class in the context of SVM? How do the support vectors relate to the hypothesis class(es)?
In classification in general, the hypothesis class is the set of possible classification functions you're considering; the learning algorithm picks a function from the hypothesis class.
For a decision tree learner, the hypothesis class would just be the set of all possible decision trees.
For a primal SVM, this is the set of functions $$\mathsf H_d =\left\{ f(x) = \operatorname{sign}\left( w^T x + b \right) \mid w \in \mathbb R^d, b \in \mathbb R \right\}.$$ The SVM learning process involves choosing a $w$ and $b$, i.e. choosing a function from this class.
For a kernelized SVM, we have some feature function $\varphi : \mathcal X \to \mathcal H$ corresponding to the kernel by $k(x, y) = \langle \varphi(x), \varphi(y) \rangle_{\mathcal H}$; here the hypothesis class becomes $$\mathsf H_k = \{ f(x) = \operatorname{sign}\left( \langle w, \varphi(x) \rangle_{\mathcal H} + b \right) \mid w \in \mathcal H, b \in \mathbb R \}.$$
Now, since $\mathcal H$ is often infinite-dimensional, we don't want to explicitly represent a $w \in \mathcal H$. But the representer theorem tells us that the $w$ which optimizes our SVM loss for a given training set $X = \{ x_i \}_{i=1}^n$ will be of the form $w = \sum_{i=1}^n \alpha_i \varphi(x_i)$. Noting that $$ \langle w, \varphi(x) \rangle_{\mathcal H} = \left\langle \sum_{i=1}^n \alpha_i \varphi(x_i), \varphi(x) \right\rangle_{\mathcal H} = \sum_{i=1}^n \alpha_i \left\langle \varphi(x_i), \varphi(x) \right\rangle_{\mathcal H} = \sum_{i=1}^n \alpha_i k(x_i, x),$$ we can thus consider only the restricted set of functions $$\mathsf H_k^X = \left\{ f(x) = \operatorname{sign}\left( \sum_{i=1}^n \alpha_i k(x_i, x) + b \right) \mid \alpha \in \mathbb R^n, b \in \mathbb R \right\}.$$ Note that $\mathsf H_k^X \subset \mathsf H_k$, but we know that the hypothesis the SVM algorithm would pick from $\mathsf H_k$ is in $\mathsf H_k^X$, so that's okay.
The support vectors specifically are the points with $\alpha_i \ne 0$. Which points are support vectors or not depends on the regularization constant and so on, so I wouldn't necessarily say that they're integrally related to the hypothesis class; but the set of possible support vectors, i.e. the training set $X$, of course defines $\mathsf H_k^X$ (along with the kernel $k$).
