Is that true that $\|g\circ f\|\leq \|g\|\cdot\|f\|$?

Solution 1:

Use the definition of norm operaror and how that translates to continuity (in the finite-dimensial case that happens always). Can you see why $$ \|g(z)||\leq ||g||||z|| ?$$ Now choose $z$ wisely and use the same argument.

Solution 2:

\begin{align} \|(gf)(x))\|=\|g(f(x)\|&\leq \|g\|\|f(x)\|\\&\le\|g\|\|f\|\|x\|\\&=(\|g\|\|f\|)\|x\|\end{align}

Now apply definition of norm of $\|gf\|$.

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Is that true that $\|g\circ f\|\leq \|g\|\cdot\|f\|$?

Solution 1:

Solution 2:

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