Extending a bilinear form (weak derivative and Bochner space)

Let $V \subset H \subset V'$ be a Hilbert triple and let $$X = \{u \in L^2(0,T;V) : u' \in L^2(0,T;H)\}$$ and $$Y = \{u \in L^2(0,T;V) : u' \in L^2(0,T;V')\}$$

Define $b:X\times X \to \mathbb{R}$ by $$b(u,v) = (u',v)_H + (v',u)_H.$$

I want to say: there is a unique extension of $b$ from $X \times X$ to $Y \times Y$ such that $$b(u,v) = \langle u', v \rangle_{V', V} + \langle v', u \rangle_{V', V}.$$

How do I show that this is true? Suppose that $X$ is dense in $Y$.

Edit: this particular $b$ is just an example. It could be something different.

-