Art of Problem Solving

User:Temperal/The Problem Solver's Resource Competition: Difference between revisions

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<cmath>
<cmath>
\begin{align*}
\begin{align*}
     |\langle (z,i), (-i,\overline{w})\rangle| &\leq ||(z,i)||||(-i,\overline{w})|| \\
     |\langle (z,1), (1,\overline{w})\rangle| &\leq ||(z,1)||||(1,\overline{w})|| \\
     |iz + iw|^2 &\leq (|z|^2 + 1)(|w|^2 + 1) \\
     |z + w|^2 &\leq (|z|^2 + 1)(|w|^2 + 1) \\
     \frac{|z + w|^2}{|z|^2 + 1} &\leq |w|^2 + 1
     \frac{|z + w|^2}{|z|^2 + 1} &\leq |w|^2 + 1
\end{align*}|
\end{align*}|
</cmath>
</cmath>

Latest revision as of 19:04, 21 August 2025

\begin{align*}     |\langle (z,1), (1,\overline{w})\rangle| &\leq ||(z,1)||||(1,\overline{w})|| \\     |z + w|^2 &\leq (|z|^2 + 1)(|w|^2 + 1) \\     \frac{|z + w|^2}{|z|^2 + 1} &\leq |w|^2 + 1 \end{align*}|