Is the zero element of a Banach space the same as the zero element of a space that it is continuously embedded within?

Question

Suppose $X$ and $Y$ are two Banach spaces and the embedding $ X \hookrightarrow Y $ is continuous. Does $ Y $ has the same zero element as $ X $? For me it seems trivial but I don't have the right arguement in hand. This question emerged from some contradiction arguement I was trying to prove that involved some $ u \in X$ such that $ \left\|u \right\|_X = 1 $ but $ \left\|u \right\|_Y = 0$ which would mean $ u = 0 $ in $Y$, hence $ \left\|u \right\|_X = 0 $ which is a contradiction. But I'm not sure how to justify that the zero element of $Y$ is the same as the zero element of $X$. We already know that since these are normed spaces, they are vector spaces. Is this enough and why? If $X$ was a subspace of $Y$ this is obvious but we do not know that.

Note: Our notion of a continuous embedding is as defined in this wikipedia article.

Answer 1

By wiki link you posed, Assuming $||x||_Y = 0 \iff x = 0_Y$ and $||x||_X = 0 \iff x = 0_X$, by continuous embedding definition you posted in Wiki link, $||0_X||_Y \leq C ||0_X||_X = 0 \implies ||0_X||_Y = 0 \implies 0_X = 0_Y$.

Answer 2

I think this can be proved with the idea of unicity of $0$. Since $X \subseteq Y$, for any $x \in X \subseteq Y$ we have $x + 0_X = x$, and I don't think this can hold for any other element of $Y$ other than $0_Y$. Maybe there can be one zero for each space if the operations of addition are defined differently, making $x_1 +_x x_2 \neq x_1 +_Y x_2$