  Consider stochastic partial differential equations (SPDEs) with fully local
monotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$:
\begin{align*} \left\{ \begin{aligned}
  dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in (0,T], \\
  X(0) & = x\in H, \end{aligned} \right. \end{align*} where \begin{align*}
  A: [0,T]\times V \rightarrow V^* , \quad B: [0,T]\times V \rightarrow
L_2(U,H) \end{align*} are measurable maps, $L_2(U,H)$ is the space of
Hilbert-Schmidt operators from $U$ to $H$ and $W$ is a $U$-cylindrical Wiener
process. Such SPDEs include many interesting models in applied fields like
fluid dynamics etc. In this paper, we establish the well-posedness of the above
SPDEs under fully local monotonicity condition solving a longstanding open
problem. The conditions on the diffusion coefficient $B(t,\cdot)$ are allowed
to depend on both the $H$-norm and $V$-norm. In the case of classical SPDEs,
this means that $B(\cdot,\cdot)$ could also depend on the gradient of the
solution. The well-posedness is obtained through a combination of
pseudo-monotonicity techniques and compactness arguments.
