What are kets?

According to Dirac’s bra-ket notation, in an inner-product space, the inner product $⟨x\mathrm{\mid }y⟩$ of vectors $x,y$ can be viewed as an application of the bra $⟨x\mathrm{\mid }$ to the ket $\mathrm{\mid }y⟩$. Here $⟨x\mathrm{\mid }$ is the linear functional $\mathrm{\mid }y⟩\mapsto ⟨x\mathrm{\mid }y⟩$ and $\mathrm{\mid }y⟩$ is the vector $y$. But often – though not always – there are advantages in seeing $\mathrm{\mid }y⟩$ as the function $a\mapsto a\cdot y$ where $a$ ranges over the scalars. For example, the outer product $\mathrm{\mid }y⟩⟨x\mathrm{\mid }$ becomes simply the composition $\mathrm{\mid }y⟩\circ ⟨x\mathrm{\mid }$. 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.