Let \(H\) denote a separable Hilbert space equipped with a Borel measure \(\nu\). The author proves, under suitable assumptions on \(\nu\), the Poincaré inequality \[ \int_H \Big | \varphi - \int_H\varphi d\nu\Big | ^2d\nu \leq C\int_H| D\varphi | ^2d\nu \] for test functions \(\varphi\) on \(H\), where \(C\) is a constant independent of \(\varphi\). This is done, initially, for \(\nu\) a Gaussian measure on \(H\), then extended to cases where \(\nu\) is absolutely continuuous with respect to a Gaussian measure on \(H\). The Poincaré inequality is then used to study the spectral gap of the infinitesimal generator of the semigroup arising from the \(H\)-valued stochastic differential equation \[ dX = (AX(t) +F(X(t)))dt + QdW \] where \(W\) is a Wiener process in \(H\), \(A\) and \(Q\) are linear operators on \(H\), and \(F: H\mapsto H\) is a nonlinear function.