Minor adjustments.

This commit is contained in:
Bokuan Li
2026-07-06 12:36:34 -04:00
parent 2cf172fa34
commit 86aba8ee4b
3 changed files with 14 additions and 2 deletions

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\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}

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\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

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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}