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Openai/6935580c-34e4-8001-9136-7bbdf1499790
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==== Consider an algebraic (noncontinuous) linear functional ℓ\ellℓ on an infinite-dimensional Banach space XXX given by choosing a Hamel basis and defining ℓ\ellℓ arbitrarily. There are sequences xn→0x_n\to 0xn→0 in norm for which ℓ(xn)\ell(x_n)ℓ(xn) does not converge to 000. If you tried to apply ℓ\ellℓ to integrals ∫fn\int f_n∫fn, nothing prevents ℓ(∫fn)\ell\big(\int f_n\big)ℓ(∫fn) from behaving pathologically. Thus algebraic linearity is insufficient. ====
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