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@@ -1,6 +1,6 @@
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database = "spec.db"
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database = "spec.db"
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document = "document.tex"
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document = "document.tex"
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siteTitle = "Garden"
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siteTitle = "Jerry's Digital Garden"
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[compiler]
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[compiler]
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compileAll = false
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compileAll = false
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@@ -9,7 +9,7 @@ indirectReferences = true
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[website]
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[website]
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font = "roboto"
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font = "roboto"
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primaryColour = "blue"
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primaryColour = "violet"
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neutralColour = "grey"
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neutralColour = "grey"
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searchLimit = 16
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searchLimit = 16
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maxSearchPages = 48
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maxSearchPages = 48
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@@ -40,7 +40,7 @@
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Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset B(E)$ be an upward-directed family that contains all finite sets, $U \subset E$ be open, and $f: U \to F$, then $f$ is \textbf{$\sigma$-differentiable on $U$} if it is $\sigma$-differentiable at every point in $U$. In which case, the map $D_\sigma f: U \to L(E; F)$ is the \textbf{$\sigma$-derivative} of $f$.
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Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset B(E)$ be an upward-directed family that contains all finite sets, $U \subset E$ be open, and $f: U \to F$, then $f$ is \textbf{$\sigma$-differentiable on $U$} if it is $\sigma$-differentiable at every point in $U$. In which case, the map $D_\sigma f: U \to L(E; F)$ is the \textbf{$\sigma$-derivative} of $f$.
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\end{definition}
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\end{definition}
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\begin{definition}]
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\begin{definition}
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\label{definition:derivative-garden}
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\label{definition:derivative-garden}
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Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma^E_{\text{Fin}}, \sigma^E_{c}, \sigma^E_b \subset 2^E$ be the collection of all finite, compact, and bounded subsets, respectively, then differentiability with respect to $\sigma^E_{\text{Fin}}, \sigma^E_{c}, \sigma^E_b$ correspond to \textbf{Gateaux}, \textbf{Hadamard}, and \textbf{Fréchet} differentiability.
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Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma^E_{\text{Fin}}, \sigma^E_{c}, \sigma^E_b \subset 2^E$ be the collection of all finite, compact, and bounded subsets, respectively, then differentiability with respect to $\sigma^E_{\text{Fin}}, \sigma^E_{c}, \sigma^E_b$ correspond to \textbf{Gateaux}, \textbf{Hadamard}, and \textbf{Fréchet} differentiability.
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\end{definition}
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\end{definition}
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