Art of Problem Solving

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

Chipdink1969 (talk | contribs)
No edit summary
Chipdink1969 (talk | contribs)
No edit summary
Line 1: Line 1:
<cmath>
<cmath>
\begin{align*}
\begin{align*}
     |\langle (z,-i), (i,\overline{w})\rangle| &\leq ||(z,-i)||||(i,\overline{w})|| \\
     |\langle (z,i), (-i,\overline{w})\rangle| &\leq ||(z,i)||||(-i,\overline{w})|| \\
     |-iz - iw|^2 &\leq (|z|^2 + 1)(|w|^2 + 1) \\
     |iz + iw|^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>

Revision as of 14:59, 8 October 2024

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