Inequality with supreme of linear bounded functions on normed vectorial spaces

Question

Let $X$ and $Y$ be normed vectorial spaces, $X\neq Y$. Let $f:X\rightarrow\mathbb{R}$ and $g:Y\rightarrow\mathbb{R}$ be linear bounded functions.

Is true the following inequality?

$$\boxed{\displaystyle\sup_{x\in X}\dfrac{f(x)}{\|x\|}+\sup_{y\in Y}\dfrac{g(y)}{\|y\|}\leq \sup_{x\in X,y\in Y}\dfrac{f(x)+g(y)}{\|x\|+\|y\|}}$$

Answer 1

No; consider $X,Y\sim \mathbb{R}$ and $f(x)=2x$, $g(x)=3x$. Then:

\begin{aligned}&&&\sup_{x\in X}\frac{f(x)}{\big\lVert x\big\rVert}=2\\ &\text{and } &&\sup_{y\in Y}\frac{g(y)}{\big\lVert y\big\rVert}=3\\ &\text{but }&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\llap{\sup_{x\in X,y\in Y}}\frac{f(x)+g(y)}{\big\lVert x\big\rVert+\big\lVert y\big\rVert}=3\end{aligned}