### Section 1 of Garrett's notes

I have put together the following brief summary of what I believe to be most important from section 1 of Garrett's notes on meromorphic continuation, "Weak-to-Strong Principles", along with some notes on the needed background and a few questions I still have about loose ends. I have TeXed it without macros, so if you cut-and-paste into your favorite editor, it should compile as an AMSLaTeX document without trouble. (If there is trouble, let me know.) This seems to be the best way of dealing with the fact that Blogger doesn't recognize any MathML.

textbf{Comments on the first section of notes, ``Meromorphic continuation

of Eisenstein Series for $\mathrm{SL}(2)$, April 2, 2001, ``Weak-to-strong

Principles"}

The main part begins with the definition of ``quasicomplete TVS"

(``TVS"=Topological vector space).

\textbf{Quasicomplete TVS}: One for which every bounded Cauchy net

is convergent.

Reminder from your favorite topology book, \textit{e.g.}, \textit{Munkres}:

\begin{itemize}

\item

a \textbf{net in $X$} is a function from a directed set $J$ into $X$.

\item $(x_{\alpha})\rightarrow x$ if $\forall U$ neighborhood

of $x$, $\exists\alpha\in J$ such that $\alpha\leq \beta\Rightarrow

x_{\beta}\in U$.

\item The definition of a Cauchy net in a TVS can then be reconstructed.

\end{itemize}

\textbf{Proposition.}\hspace*{0.3cm} Continuous compactly supported function $f: X\rightarrow V$

with values in a \textbf{quasicomplete} (locally convex) TVS $V$

have GP [=Gelfand-Pettis=vector-valued] integrals with respect to finite

positive regular Borel measures $\mu$ on compact spaces $X$,

and these integrals are unique.

\begin{proof}\textit{Overview of the Proof.} The proof has two parts.

\begin{itemize}

\item[*] $\forall K\subset V$,

$\overline{\mathrm{Co}(K)}$ is compact.

\end{itemize}

\begin{itemize}

\item[(i)] If $V$ has Property $(*)$, then the conclusions hold. This

is shown, essentially, in the proof Theorem 3.27 of Rudin, although the statement

of the Theorem is less general.

\item[(ii)] Every quasi-complete space has Property $(*)$. I take

it this follows

from the strengthened form of the Krein-Millman theorem found as

Theorem 3, $\S \mathrm{IV}.5$ of Bourbaki's \textit{TVS's}, but

I have not checked this point in detail.

\end{itemize}

\end{proof}

Remembering the following formula is useful for differentiating under

the GP integral sign. Under suitable hypotheses (stated precisely

by Garrett) which will always be valid in our applications we have

\[

T\left(\int_X f(x)\,\text{d}\mu(x)\right)=\int_X Tf(x)\, \text{d}\mu(x).

\]

Here,

\begin{itemize}

\item $T: V\rightarrow W$ (continuous linear map).

\item $f: X\rightarrow V$ (continuous compactly supported $V$-valued

function).

\item $\mu$ (finite positive Borel measure).

\end{itemize}

\textbf{Corollary} (of the first proposition) Let $V$ be a quasicomplete

(and locally convex) TVS. Then weakly holomorphic $V$-valued

functions are strongly holmorphic.

\begin{proof} The pattern for the proof is the same as that

of part (c) of Rudin's Theorem 3.31 from part (b) of the same theorem.

The only difference is that we have weakened his hypothesis that $V$

(``$X$") is a complete Fr\'{e}chet space; used here, is the fact that

in Garrett's result on the existence of GP integrals, we have used

only one property of $V$: quasicompleteness.

\end{proof}

\textit{The next corollary seems to belong after a ``proposition" which doesn't

exist in the text.} It would be nice to have a reference at least,

but Garrett provides none, and I am so far unable to find one either.

In order to understand this corollary, we need to recall what a topology

induced from seminorms is.

\textbf{Theorem 1.37} Suppose $\mathscr{P}$ is a separating

($\forall x\neq 0$, $\exists p\in \mathscr{P}$ with $p(x)\neq 0$)

family of seminorms on a vector space $X$. Associate to each

$p\in\mathscr{P}$ and to each positive integer $n$

the set

\[

V(p,n)=\{x:\, p(x)<1/n\}.

\]

Let $\mathscr{B}$ be the collection of all finite intersections of the

sets $V(p,n)$. Then $\mathscr{B}$ is a convex balanced local base

for a topology $\tau$ on $X$, which turns $X$ into a locally convex

vector space such that

\begin{itemize}

\item[(a)] every $p\in\mathscr{P}$ is continuous.

\item[(b)] a set $E\subset X$ is bounded if and only if every

$p\in\mathscr{P}$ is bounded on $E$.

\end{itemize}

\textit{Question/Exercise}: Is the topology induced on $\mathrm{Hom}^0(X,Y)$

\textit{by the family of seminorms $\rho_{x,\mu}(T)=|\mu(T(x))|$}

the same as the weak*-topology?

\textbf{Corollary.}\hspace*{0.5mm} A weakly holomorphic (means

all $\mathbf{C}$-valued function $s\mapsto\mu(T_S(x)))$ are holomorphic)

$\mathrm{Hom}^0(X,Y)$-valued function (where $X$ is an LF-space

and $Y$ is a quasicompact space) is (strongly) holomorphic.

Here, (strongly) holomorphic refers to the existence of the limit

in the weak operator topology induced by the family of seminorms

\[

\rho_{x,\mu}(T)=|\mu(T(x))|.

\]

textbf{Comments on the first section of notes, ``Meromorphic continuation

of Eisenstein Series for $\mathrm{SL}(2)$, April 2, 2001, ``Weak-to-strong

Principles"}

The main part begins with the definition of ``quasicomplete TVS"

(``TVS"=Topological vector space).

\textbf{Quasicomplete TVS}: One for which every bounded Cauchy net

is convergent.

Reminder from your favorite topology book, \textit{e.g.}, \textit{Munkres}:

\begin{itemize}

\item

a \textbf{net in $X$} is a function from a directed set $J$ into $X$.

\item $(x_{\alpha})\rightarrow x$ if $\forall U$ neighborhood

of $x$, $\exists\alpha\in J$ such that $\alpha\leq \beta\Rightarrow

x_{\beta}\in U$.

\item The definition of a Cauchy net in a TVS can then be reconstructed.

\end{itemize}

\textbf{Proposition.}\hspace*{0.3cm} Continuous compactly supported function $f: X\rightarrow V$

with values in a \textbf{quasicomplete} (locally convex) TVS $V$

have GP [=Gelfand-Pettis=vector-valued] integrals with respect to finite

positive regular Borel measures $\mu$ on compact spaces $X$,

and these integrals are unique.

\begin{proof}\textit{Overview of the Proof.} The proof has two parts.

\begin{itemize}

\item[*] $\forall K\subset V$,

$\overline{\mathrm{Co}(K)}$ is compact.

\end{itemize}

\begin{itemize}

\item[(i)] If $V$ has Property $(*)$, then the conclusions hold. This

is shown, essentially, in the proof Theorem 3.27 of Rudin, although the statement

of the Theorem is less general.

\item[(ii)] Every quasi-complete space has Property $(*)$. I take

it this follows

from the strengthened form of the Krein-Millman theorem found as

Theorem 3, $\S \mathrm{IV}.5$ of Bourbaki's \textit{TVS's}, but

I have not checked this point in detail.

\end{itemize}

\end{proof}

Remembering the following formula is useful for differentiating under

the GP integral sign. Under suitable hypotheses (stated precisely

by Garrett) which will always be valid in our applications we have

\[

T\left(\int_X f(x)\,\text{d}\mu(x)\right)=\int_X Tf(x)\, \text{d}\mu(x).

\]

Here,

\begin{itemize}

\item $T: V\rightarrow W$ (continuous linear map).

\item $f: X\rightarrow V$ (continuous compactly supported $V$-valued

function).

\item $\mu$ (finite positive Borel measure).

\end{itemize}

\textbf{Corollary} (of the first proposition) Let $V$ be a quasicomplete

(and locally convex) TVS. Then weakly holomorphic $V$-valued

functions are strongly holmorphic.

\begin{proof} The pattern for the proof is the same as that

of part (c) of Rudin's Theorem 3.31 from part (b) of the same theorem.

The only difference is that we have weakened his hypothesis that $V$

(``$X$") is a complete Fr\'{e}chet space; used here, is the fact that

in Garrett's result on the existence of GP integrals, we have used

only one property of $V$: quasicompleteness.

\end{proof}

\textit{The next corollary seems to belong after a ``proposition" which doesn't

exist in the text.} It would be nice to have a reference at least,

but Garrett provides none, and I am so far unable to find one either.

In order to understand this corollary, we need to recall what a topology

induced from seminorms is.

\textbf{Theorem 1.37} Suppose $\mathscr{P}$ is a separating

($\forall x\neq 0$, $\exists p\in \mathscr{P}$ with $p(x)\neq 0$)

family of seminorms on a vector space $X$. Associate to each

$p\in\mathscr{P}$ and to each positive integer $n$

the set

\[

V(p,n)=\{x:\, p(x)<1/n\}.

\]

Let $\mathscr{B}$ be the collection of all finite intersections of the

sets $V(p,n)$. Then $\mathscr{B}$ is a convex balanced local base

for a topology $\tau$ on $X$, which turns $X$ into a locally convex

vector space such that

\begin{itemize}

\item[(a)] every $p\in\mathscr{P}$ is continuous.

\item[(b)] a set $E\subset X$ is bounded if and only if every

$p\in\mathscr{P}$ is bounded on $E$.

\end{itemize}

\textit{Question/Exercise}: Is the topology induced on $\mathrm{Hom}^0(X,Y)$

\textit{by the family of seminorms $\rho_{x,\mu}(T)=|\mu(T(x))|$}

the same as the weak*-topology?

\textbf{Corollary.}\hspace*{0.5mm} A weakly holomorphic (means

all $\mathbf{C}$-valued function $s\mapsto\mu(T_S(x)))$ are holomorphic)

$\mathrm{Hom}^0(X,Y)$-valued function (where $X$ is an LF-space

and $Y$ is a quasicompact space) is (strongly) holomorphic.

Here, (strongly) holomorphic refers to the existence of the limit

in the weak operator topology induced by the family of seminorms

\[

\rho_{x,\mu}(T)=|\mu(T(x))|.

\]

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