The following definition of convex conjugate is taken from Wiki:

Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$ Denote the dual pairing by $$\langle \cdot ,\cdot \rangle :X^{*}\times X\to \mathbb {R}.$$ For a function $f:X\to\mathbb{R}\cup \{\pm\infty\}$taking values on the extended real number line, the

convex conjugate$$f^{*}:X^{*}\to \mathbb {R} \cup \{-\infty ,+\infty \}$$ is defined in terms of the supremum by $$f^{*}\left(x^{*}\right):=\sup \left\{\left.\left\langle x^{*},x\right\rangle -f\left(x\right)\right|x\in X\right\}.$$

Bachir introduced the notion conjugate $f^\times$ of $f:X\to\mathbb{R}$ as $$f^\times (\phi) := \sup_{x\in X}\{\phi(x) - f(x)\}$$ for all $\phi\in C_b(X),$ the set of all bounded real-valued continuous functions on $X.$

Question:Do we have convex conjugate for vector-valued function? More precisely, if $E$ is a Banach space, can we define convex conjugate of $f:X\to E$ by $$\tilde{f}^\times(\phi) = \sup_{x\in X}\{\|\phi(x)\| - \|f(x)\|\}$$ for all $\phi\in C_b(X,E),$ the set of all bounded $E$-valued continuous functions on $X?$