The discussion confirms that the triangle inequality is applicable to norms of integral operators. Specifically, for any two integral operators \( K \) and \( L \) defined by the equation \( (Ky)(x)=\int_{a}^{b} k(x,s)y(s)ds \), the relationship \( ||L|| + ||K-L|| \ge ||K|| \) holds true. This conclusion is derived directly from the properties of norms and the triangle inequality, establishing a foundational principle in functional analysis.
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That follows from the triangle inequality: $\|K\| = \|K-L+L\| \leqslant \|K-L\| + \|L\|.$sarrah said:Can I always say without reservation that for any two integral operators $K$ and $L$ defined as follows say
$(Ky)(x)=\int_{a}^{b} \,k(x,s)y(s)ds$
that
$||L||+||K-L||\ge||K||$
thanks
Sarrah