Say I have a $\beta$-contraction $T$, e.g., $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $|| T(x) - T(y)|| < \beta ||x-y||$, $\beta <1$. Now, suppose I want to work with another norm $\| \|_{2}$. Under this norm, will $T$ still be a contraction? Thanks a lot!
2026-03-25 19:00:24.1774465224
Changing the norm will affect a contraction?
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In general, no, it might not be a contraction. A sufficient condition for it to be a contraction would be $$\alpha \| \cdot \|_2 \le \| \cdot \| \le \gamma \|\cdot\|_2$$ where $\beta \gamma < \alpha$.