Remark 2

Let $j$ be the canonical inclusion of Sp($u$) in $\mathbf{C}$. Since Sp($u$) is closed, the mapping $j$ is $\mu$-proper for every bounded measure $\mu$ on Sp($u$) (INT, V, p. 68, § 6, n$^o1$, Proposition 1). One will often identify the spectral measures of $u$ with the measures on $\mathbf{C}$ which are their images under $j($cf. INT, V, p. 84, § 7, n$^o2)$.

The mapping of $E\times E$ into the Banach space $\mathscr{M}^1$(Sp($u$)) which associates to $(x, y)$ the spectral measure of $(x, y)$ relative to $u$ is sesquilinear and continuous of norm $\leqslant 1$.