What are kets?

According to Dirac's bra-ket notation, in an inner-product space, the inner product ⟨ x | y⟩ of vectors x,y can be viewed as an application of the bra ⟨ x| to the ket |y⟩. Here ⟨ x| is the linear functional |y⟩↦⟨ x | y⟩ and |y⟩ is the vector y. But often – though not always – there are advantages in seeing |y⟩ as the function a ↦ a· y where a ranges over the scalars. For example, the outer product |y⟩⟨ x| becomes simply the composition |y⟩∘⟨ x|. It would be most convenient to view kets sometimes as vectors and sometimes as functions, depending on the context. This turns out to be possible. While the bra-ket notation arose in quantum mechanics, this note presupposes no familiarity with quantum mechanics.