Norm of Volterra Operator with $L^1$
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I am trying to find the Operator norm for the following operator $$(Vf)(t):(C[0,1],L^1)to (C[0,1],L^1)$$ $$fmapsto int_0^t f(s)ds$$ What I have done $$|Vf(t)|=sup_{|f|_{L^1}=1}{|int_0^t f(s) |ds}_{L^1}=sup_{|{f}|_{L^1}=1}int_0^1|{int_0^t f(s)| ~ds}dtleq sup_{|{f}|_{L^1}=1}int_0^1int_0^t |{f(s)}|~dsdt .$$ Tonelli: $$=sup_{|{f}|_{L^1}=1}int_0^tint_0^1 |f(s)|~ dt ds =sup_{|{f}|_{L^1}=1}int_0^t |f(s)|~ds leq int_0^1|f(s)|~ds =|{f}|_{L^1}=1 $$ this doesnt seem all too correct. How do I continue?
functional-analysis
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asked Nov 27 '18 at 22:10
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