Let $Y\subset X$ be Banach spaces, then is $X+Y\subset X$ a continuous embedding?

Question

Define the norm $$\lVert x\rVert_{X+Y}=\inf_{a+b=x}(\lVert a\rVert_X+\lVert b\rVert_Y)$$ on $X+Y$, then is it true that $X+Y\subset X$ embeds continuously?

Answer 1

It seems to me that $X+Y \subset X$ embeds continuously if and only if $Y \subset X$ embeds continuously. Indeed suppose that ${\left\|\cdot \right\|}_{X} \leqslant C {\left\|\cdot \right\|}_{Y}$, then if $x = a+b$, one has

$${\left\|x\right\|}_{X} = {\left\|a+b\right\|}_{X} \leqslant {\left\|a\right\|}_{X}+{\left\|b\right\|}_{X} \leqslant {\left\|a\right\|}_{X}+C {\left\|b\right\|}_{Y} \leqslant \left(1+C\right) \left({\left\|a\right\|}_{X}+{\left\|b\right\|}_{Y}\right)$$

taking the infimum over $a \in X$ and $b \in Y$ yields

$${\left\|x\right\|}_{X} \leqslant \left(1+C\right) {\left\|x\right\|}_{X+Y}$$

Conversely, suppose that ${\left\|\cdot \right\|}_{X} \leqslant C {\left\|\cdot \right\|}_{X+Y}$, then for any $y \in Y$, one can write $y = 0+y$, hence

$${\left\|y\right\|}_{Y} = {\left\|0\right\|}_{X}+{\left\|y\right\|}_{Y} \in \left\{{\left\|a\right\|}_{X}+{\left\|b\right\|}_{Y} , y = a+b , a \in X , b \in Y\right\}$$

which implies ${\left\|y\right\|}_{Y} \geqslant {\left\|y\right\|}_{X+Y}$. It follows that

$${\left\|y\right\|}_{X} \leqslant C {\left\|y\right\|}_{X+Y} \leqslant C {\left\|y\right\|}_{Y}$$

Now the question: can you find an example with a non continuous embedding of $Y$?