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Fix typo.
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Bokuan Li
2026-07-09 12:47:54 -04:00
parent 013b095fa2
commit 67b00db276
1 changed files with 2 additions and 2 deletions
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src/fa/lc/hahn-banach.tex
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@@ -26,10 +26,10 @@
then for any $x \in F$ and $t > 0$, then for any $x \in F$ and $t > 0$,
\begin{align*} \begin{align*}
\phi(x + tx_0) &= \phi(x) + t\lambda = t\braks{\phi(t^{-1}x) + \lambda} \\ \phi_{x_0, \lambda}(x + tx_0) &= \phi(x) + t\lambda = t\braks{\phi(t^{-1}x) + \lambda} \\
&\le t\braks{\rho(t^{-1}x + x_0) - \phi(t^{-1}x) + \phi(t^{-1}x)} \\ &\le t\braks{\rho(t^{-1}x + x_0) - \phi(t^{-1}x) + \phi(t^{-1}x)} \\
&= t\rho(t^{-1}x + x_0) = \rho(x + tx_0) \\ &= t\rho(t^{-1}x + x_0) = \rho(x + tx_0) \\
\phi(x - tx_0) &= \phi(x) - t\lambda = t\braks{\phi(t^{-1}x) - \lambda} \\ \phi_{x_0, \lambda}(x - tx_0) &= \phi(x) - t\lambda = t\braks{\phi(t^{-1}x) - \lambda} \\
&\ge t\braks{\rho(t^{-1}x - x_0) + \phi(t^{-1}x) - \phi(t^{-1}x)} \\ &\ge t\braks{\rho(t^{-1}x - x_0) + \phi(t^{-1}x) - \phi(t^{-1}x)} \\
&= t\rho(t^{-1}x - x_0) = \rho(x - tx_0) &= t\rho(t^{-1}x - x_0) = \rho(x - tx_0)
\end{align*} \end{align*}