FA24

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Let L^{2}[a,b]=\left\{f:\int _{a}^{b}\left|f(t)\right|^{2}dt<\infty \right\}\, and define ||f||=\left(\int _{a}^{b}\left|f(t)\right|^{2}dt\right)^{{1/2}}\,. Show that

(Tf)(t)=\int _{a}^{b}K(s,t)f(s)ds\,

defines a bounded linear operator on L^{2}[a,b]\, when K(s,t)\, is a continuous function on [a,b]\times [a,b]\,. Estimate the norm of T\,.