Remark 1

Suppose the hilbertian space E does not reduce to 0. We identify $\mathscr{L}(K;E)$ with E by the mapping $u \mapsto u(1)$; in other words, the vector x of E is identified with the mapping $\lambda \mapsto \lambda.x$ from K into E. Then the adjoint of x is the mapping $x^*:E \to K$ given by $x^*(y) = \langle x|y \rangle$. In other words, $x \mapsto x^*$ is the canonical semi-linear mapping from E onto its dual (V, p. 15).

Similarly, we identify the number $\lambda \in K$ with the endomorphism $\lambda.1_E$ of E. Then $\lambda^*$ is precisely the conjugate of $\lambda$.

With these identifications, we can define a product $t_1 ... t_n$ where each $t_i$ is, either a number in K, or a vector in E, or a linear form belonging to E', or an element of $\mathscr{L}(E)$, provided that there are never two consecutive factors $t_i$ and $t_{i+1}$ of one of the following types : • xy where x, y are both in E, or both in E'; • xA or Ax' with $A \in \mathscr{L}(E)$, $x \in E$ and $x' \in E'$.

We have the following rules of composition : a) associativity ; b) every element of K commutes with all the other factors ; c) we have $(t_1 ... t_n)^* = t_n^* ... t_1^*$; in other words, the adjoint of a product is the product of the adjoints taken in the reverse order. Also $t^{**} = t$.

For example, let x, y be in E and let A be in $\mathscr{L}(E)$. Then $x^*y$ represents the scalar product $\langle x|y \rangle$ and $x^*Ay$ represents the scalar product $\langle x|Ay \rangle$. We also have $(A^*x)^* = x^*A^{**} = x^*A$, hence $(A^*x)^*y = x^*Ay$, which can be interpreted as

in conformity with the definition of the adjoint. We observe that $yx^*$ is the endomorphism $z \mapsto y \langle x|z \rangle$ of E, since $yx^*z$ can be interpreted as $y(x^*z)$ by associativity, or as $y.\langle x|z \rangle$.

Following Dirac $^1$, in most works of Mathematical Physics, the elements of E are represented by the symbol $|x\rangle$, those of E' by $\langle t|$. The scalar product is written as $\langle x|y \rangle = \langle x|.|y \rangle$ and the first rule of interdiction in the products excludes the combinations of the signs $|$ and $|<$, for example $|x\rangle|y\rangle$.