  We study the notions of mild solution and generalized solution to a linear
stochastic partial differential equation driven by a pure jump symmetric L\'evy
white noise. We identify conditions for existence for these two kinds of
solutions, and we identify conditions under which they are essentially
equivalent. We establish a necessary condition for the existence of a random
field solution to a linear SPDE, and we apply this result to the linear
stochastic heat, wave and Poisson equations driven by a symmetric
$\alpha$-stable noise.
