Let X be an Banach Algebra. Suppose $M_n\rightarrow M$ in norm topology. Show that lim sup$\sigma(M_n)\le \sigma(M)$.
By a theorem of Gelfand, $\sigma(M)=lim_{k\rightarrow\infty} |M^k|^{\frac{1}{k}}$. Do I need to use this formula?
2026-03-27 08:29:48.1774600188
Upper Semicontinuity of Spectral Radius
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