In space $C[0,1]$ is defined bounded and compact operator $A$: $$ Ax(t)= 2x(0) - tx(1),\enspace t\in [0,1] $$ Find the norm of operator $A$. ($\|A\|=$ ?)
2026-04-08 23:05:47.1775689547
Find the norm of a bounded compact operator
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The norm of $A$ is $3$. $$ |Ax(t)|\leq 2|x(0)|+t\,|x(1)|\leq 2\|x\|+\|x\|=3\|x\|. $$ So $\|A\|\leq 3$. Now let $x(t)=1-2t$. The range of $x$ is $[-1,1]$, so $\|x\|=1$. And $$ |Ax(1)|=|2x(0)-x(1)|=2-(-1)=3. $$ Thus $\|A\|\geq3$, showing the equality.