I was not formally introduced to this math, so I appreciate corrections but I'll give my impressions. The fock space for a particle with space H is (c, H, HxH, HxHxH, ... ) 1: What is a hilbert space for a single particle? I believe know what a hilbert space is, but for 'a particle'.. what is the base of this space? what to its vectors mean? 2: Operaions on Direct sums of spaces are defined as parallel operations over vector of those spaces. If each position on a tuple does not have a single vector associated(or it has?), like a basis vector, what would mean in the fock space, for example, (0, 1 ,1 , 0 ..0..), (0, 2 ,1 , 0 ..0..), (0, 1 ,1 , 0 ..0..) or (0, 0.5 ,0 , 0 ..0..) 3: How would fock states be denoted as tuples of the fock space? like, would |2> be (0,0,1,0...) ? 4: What does it mean for a state to have a 'well defined number of particles'? that all except one coefficient on a fock tuple is non zero?