Let assume the following linear operator $$ A: X \to X $$ $$ Ax(s)=\int_{0}^{1} k(s,t) x(t)\,\mathrm{d}t $$

Where $X=C([0,1], \|.\|_{\infty})$ and $k(s,t)$ is a continuous function in $[0,1]^2$.

It is easy to prove that the operator norm, $\|A\|$, satisfies $$ \|A\| \leq \max_{0 \leq s \leq 1} \int_{0}^{1} |k(s,t)| \,\mathrm{d}t \,\,\, (●)$$

I have asked to prove that if $k$ is nonnegative then we actually have equality in $(●)$.

Can some one give an example that the inequality is strict.



Actually the inequality is strict for all $k$ coninuous in $[0,1]$.

Take for example $$x_{\varepsilon}(t) = \dfrac{k(s_0,t)}{|k(s_0,t)|+\varepsilon}$$ where $s_0$ is the point where $\int^1_0 |k(s,t)| dt$ attains its maximum ($s_0$ exists since $k$ is continuous).

An estimate for all $\varepsilon >0$ with this function should give you the equality.


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