1 Linear vectors and Hilbert space D: Linear vector space A linear vector space is a set of elements, called vectors, which is closed under addition and multiplication by scalars.

Using Dirac notation, the vectors are denoted by kets: |k). We can associate to each ket a vector in the dual space called bra: (ψ|.

If two vectors |ψ) and |ϕ) are part of a vector space, then ψ + ϕ also belongs to the space. If a vector ψ is in the space, then α |ψ) is also in the space (where α is a complex scalar).

A set of linearly independent vectors ϕi is such that k ck ϕk = 0 if and only if ck = 0 k (no trivial combination of them sums to zero).