2026-03-25 12:21:12.1774441272
Proof of : Norm of Linear transformation from Rn (equipped with sup-norm) to Rm (equipped with l1 norm)
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The inequality says $\sum_{i=1}^{m} |\sum_{k=1}^{n} b_{ik} x_k| \leq \sum _{i,j} |b_{ij}| \sup\{|x_k|1 \leq k \leq n\}$ whose proof is obvious. Here $(b_{ij})$ is the matrix of the LT A. The norm of a LT A is defined by $||A||=\sup \{||Ax||:||x|| \leq 1\}$. The inequality I just stated says that $||Ax|| \leq \sum _{i,j} |b_{ij}| ||x||$.