I have a textbook (Nonlinear Spectral Theory, Appell et al 2004) that gives the following definitions and properties.
Let $X$ and $Y$ be Banach spaces, and $f:X\rightarrow Y$ a continuous (possibly nonlinear) operator. Define the Lipschitz seminorms as $$[f]_{Lip} = \sup_{x\neq y} \frac{||f(x)-f(y)||}{||x-y||}$$ $$[f]_{lip} = \inf_{x\neq y} \frac{||f(x)-f(y)||}{||x-y||}$$
The book then gives the following property for two continuous operators $f$ and $g$: $$[f]_{lip}[g]_{lip}\leq [fg]_{Lip}\leq[f]_{lip}[g]_{Lip}$$ Why is the second inequality true? I can see why we would have submultiplicativity with $ [fg]_{Lip}\leq[f]_{Lip}[g]_{Lip}$ but don't understand the stronger inequality that is given. The book says that the inequalities "are straightforward consequences of calculation rules for suprema and infima" but I'm afraid it's not straightforward for me.
Thanks in advance for your help! It would be very useful if I could apply this result, but I don't want to use something that I don't understand.
There may be a typo. Take $X$ and $Y$ to be two dimensional and $f:X\to Y$ linear with matrix $\begin{pmatrix}1&0 \\ 0&0.5 \end{pmatrix}$ and $g:Y\to X$ also linear with same matrix. Then $[f]_{lip}=0.5$, $[g]_{Lip}=1$ and $[fg]_{Lip}=1$ so we would have $1\leq 0.5$. So this is not true.