In this chapter, we will more formally discuss the bra and ket notation introduced in the previous chapter.
Let be a Hilbert space and let be a continuous linear transformation. Then is said to be a linear functional on .
The space of all linear functionals on is denoted as . Notice that is a normed vector space on with
We also have the obvious definition, are said to be orthogonal if . We write this as . If then we write if
Let be a Hilbert space, let be a closed subspace of and let . Then, every can be written where
Reisz representation theorem
Let be a Hilbert space. Then, every (that is is a linear functional) can be expressed as an inner product.