fa.functional analysis - Unbounded operator bounded in a dense subset

X is infinite-dimensional, so we can find $(e_n)$ a linearly independent sequence in X; let X' be the span. By rescaling, we can assume that $|e_n|=1$ for each n.
Define $T:X'rightarrow mathbb R$ (or $mathbb C$, or embed into Y if you wish) by $T(e_n) = n$ for each n. Clearly T is unbounded on X'.

For each finite sum $x=sum_{n=1}^N x_n e_n$ and $epsilon>0$, we can choose $ainmathbb R$ and $m$ very large with $|a|<epsilon$ and $T(x) = -am$. Set $y=x+ae_m$, so $T(y)=T(x)+am=0$ and also $|x-y| = |a|<epsilon$. Let D' be the collection of all such $y$; as such $x$ exhaust X', we certainly have that D' is dense in X'.

Use Zorn to extend $E={e_n}$ to $E'$ a basis of X. Extend T to X by setting $T(x)=0$ for $xin E' setminus E$. Let $D = D' + text{span}(E'setminus E)$, so D is dense in X, and T is bounded on D; actually T vanishes on D.

Now, D is certainly not a subspace: if you want that as well, I don't know!