Fixed wrong proof marker.
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.vscode/project.code-snippets
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.vscode/project.code-snippets
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@@ -175,6 +175,6 @@
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"Cite Proof": {
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"Cite Proof": {
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"scope": "latex",
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"scope": "latex",
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"prefix": "cproof",
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"prefix": "cproof",
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"body": ["{Proof, {{\\cite[$1]{$2}}}. }$0"]
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"body": ["[Proof, {{\\cite[$1]{$2}}}. ]$0"]
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}
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}
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}
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}
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@@ -17,7 +17,7 @@
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is a homeomorphism. Under the identification $\beta X = \Omega(BC(X; \complex))$, $\Gamma_{BC(X; \complex)} = \beta$.
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is a homeomorphism. Under the identification $\beta X = \Omega(BC(X; \complex))$, $\Gamma_{BC(X; \complex)} = \beta$.
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\end{theorem}
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\end{theorem}
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\begin{proof}{Proof, {{\cite[Theorem I.6.4]{Zhu}}}. }
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\begin{proof}[Proof, {{\cite[Theorem I.6.4]{Zhu}}}. ]
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Let $\phi \in BC(X; \complex)^* \setminus \ol{E(X)}$, then there exists $\seqf{f_k} \subset BC(X; \complex)$ and $\eps > 0$ such that for every $x \in X$,
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Let $\phi \in BC(X; \complex)^* \setminus \ol{E(X)}$, then there exists $\seqf{f_k} \subset BC(X; \complex)$ and $\eps > 0$ such that for every $x \in X$,
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\[
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\[
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f(x) = \sum_{k = 1}^n |f_k(x) - \dpn{f_k, \phi}{BC(X; \complex)}|^2 \ge \eps^2
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f(x) = \sum_{k = 1}^n |f_k(x) - \dpn{f_k, \phi}{BC(X; \complex)}|^2 \ge \eps^2
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@@ -58,7 +58,7 @@
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\Gamma(f)(z) = \sum_{n \in \integer} f(n)z^n
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\Gamma(f)(z) = \sum_{n \in \integer} f(n)z^n
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\]
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\]
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\end{theorem}
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\end{theorem}
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\begin{proof}{Proof, {{\cite[Theorem 6.3]{Zhu}}}. }
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\begin{proof}[Proof, {{\cite[Theorem 6.3]{Zhu}}}. ]
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Let $z \in \mathbf{S}^1$ and $f, g \in \ell^1(\integer)$, then by \hyperref[Fubini's Theorem]{theorem:fubini-tonelli},
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Let $z \in \mathbf{S}^1$ and $f, g \in \ell^1(\integer)$, then by \hyperref[Fubini's Theorem]{theorem:fubini-tonelli},
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\begin{align*}
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\begin{align*}
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F(z)(f * g) &= \sum_{n \in \integer}z^n \sum_{k \in \integer}f(k)g(n-k) \\
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F(z)(f * g) &= \sum_{n \in \integer}z^n \sum_{k \in \integer}f(k)g(n-k) \\
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