Applying Sion's Minimax Theorem for Simple Linear Function

Question

I want to know whether Sion's Minimax Theorem is applicable to the following instance: \begin{align*} \max_{x\in \mathbb{R}^n} \min_{w\in S} v(x,w) &:= \max_{x\in \mathbb{R}^n} \min_{w\in S}\left\{x^Td -c^Tw + x^T w\right\}\\ &= \min_{w\in S}\max_{x\in \mathbb{R}^n} \left\{x^T d -c^Tw + x^T w\right\} \tag{1} \end{align*} where $S\subset \mathbb{R}^n$ is a compact and convex set and $c,d$ are fixed vectors. Can we consider $\mathbb{R}^n$ to be a convex subset of topological vector space and get the equality as in (1) using Sion's Theorem? I am new to concepts in topology and minimax problems and being confused because (1) can be written as $$ \min_{w\in S}\{\max_{x\in \mathbb{R}^n} \left\{x^T d + x^T w\} -c^Tw \right\} $$
and the inner bracket goes to infinity. Does it mean that the optimal value is positive infinity?

Answer 1

Well since $x$ does not live in a compact set I don't see how Sion's minimax theorem would guarantee equality.