I am struggeling to understand the definition for the convex conjugate as stated by wiki: Let $X$ be a real topologacl vector space, and let $X^*$ be the dual space to $X$. Denote the dual pairing by $$\langle \cdot , \cdot \rangle : X^* \times X \longrightarrow \mathbb{R}. $$
For a function $$f : X \longrightarrow \mathbb{R} \cup \{ -\infty, \infty \}$$ the convex conjugate $$f^*: X^*\longrightarrow \mathbb{R} \cup \{ -\infty, \infty \} $$ is defined in terms of the supremum by $$ f^*(x^*) := \sup\{\langle x^*, x \rangle - f(x) \ | \ x \in X \} .$$
From what I can gather, the dual space to $X$ consists of all $\textit{linear functionals}$ $X \mapsto \mathbb{R}$. However, I fail to see how this applies to the definiton given above.
If we took $X = \mathbb{R^2}$, a linear functional on $\mathbb{R}^2$ would be $\varphi_{y}(x) = \langle x, y\rangle$. This makes it seem like the elements from $X^*$ above are rather elements which make $\langle \cdot, \cdot \rangle$ a linear functional. It does not seem to me like $$f^*(\varphi_y) = \sup\{\langle \varphi_y, x\rangle - f(x)\ | \ x \in \mathbb{R^2} \} $$ for some $y \in \mathbb{R}^2$ makes much sense.
Is there something about the definition of dual spaces I have misunderstood? Or should this be understood as $X^*$ is the space of all $y$ such that $\langle y, \cdot \rangle$ is a linear functional on $X$?