diff --git a/src/fa/tvs/complexify.tex b/src/fa/tvs/complexify.tex index bd00c2b..4fd7013 100644 --- a/src/fa/tvs/complexify.tex +++ b/src/fa/tvs/complexify.tex @@ -84,6 +84,18 @@ \end{proof} +\begin{definition}[Hermitian] +\label{definition:hermitian-functional} + Let $E$ be a vector space over $\real$, $*: \complex(E) \to \complex(E)$ be the canonical complex conjugation map, and $\phi \in \hom(\complex(E); \complex)$, then the following are equivalent: + \begin{enumerate} + \item $\phi|_E \in \hom(E; \real)$. + \item For each $x \in E$, $\dpn{x, \phi}{\complex(E)} = \ol{\dpn{x^*, \phi}{\complex(E)}}$. + \end{enumerate} + + If the above holds, then $\phi$ is \textbf{Hermitian}. +\end{definition} + + diff --git a/src/measure/measure/scaffold.tex b/src/measure/measure/scaffold.tex index b7541fd..4fb139a 100644 --- a/src/measure/measure/scaffold.tex +++ b/src/measure/measure/scaffold.tex @@ -16,7 +16,7 @@ \end{definition} \begin{example} - Let $X$ be a LCH space, $\mu$ be a Radon measure, and $\mathcal{K}$ be the collection of compact subsets of $X$, then $\mathcal{K}$ is a scaffold for $\mu$. + Let $X$ be a LCH space, $\mu$ be a semifinite Radon measure, and $\mathcal{K}$ be the collection of compact subsets of $X$, then $\mathcal{K}$ is a scaffold for $\mu$. \end{example} % Omitted diff --git a/src/op/c-star/homomorphism.tex b/src/op/c-star/homomorphism.tex index 33a717d..3dfd820 100644 --- a/src/op/c-star/homomorphism.tex +++ b/src/op/c-star/homomorphism.tex @@ -12,7 +12,7 @@ If $A$ and $B$ are unital, then $\phi$ is \textbf{unital} if: \begin{enumerate} - \item[(U)] $\phi(1_A) = \one_B$. + \item[(U)] $\phi(1_A) = 1_B$. \end{enumerate} \end{definition}