Let X be a complex Banach space and denote by \(F^p_\alpha (X)\) and \(F^{w,p}_\alpha (X)\) the X-valued Fock spaces of entire functions f such that \(\Vert f(z)\Vert e^{-\frac{\alpha }{2}|z|^2}\in L^p(dA)\) and \(x^*(f(z))e^{-\frac{\alpha }{2}|z|^2}\in L^p(dA)\) for any \(x^*\in X^*\) , respectively. For Hilbert-valued functions, it is shown that \(F^2_\alpha (H)=F^{w,2}_\alpha (H)\) if and only if H is finite dimensional and also that \(F^2_\alpha (H)\) can be identified with the space of Hilbert-Schmidt operators from \(H\rightarrow F^2_\alpha \) . A proof of the density of polynomials in \(F^p_\alpha (X)\) for \(p<\infty \) is presented and also the expected duality \((F^p_\alpha (X))^*= F^q_\alpha (X^*)\) with \(1\le p<\infty \) and \(1/p+1/q=1\) is provided.
