Let $B$ a uniformly convex Banach space and let $A$ be continuously embedded in $B$, i.e. $$A\hookrightarrow B,$$ which means that $A\subseteq B$ and there exists $c>0$ such that $\|u\|_B\le C\|u\|_A$ for any $u\in A$.
The question is that the continuous embedding is enough to say that $A$ is a uniformly Banach space itself.
I do not how to prove this property because I find my definition of uniform convexity not so much "usable". The definition was stated as: A Banach space is uniformly convex if and only if its dual is uniformly smooth.
Anyone could please help me with that?
This is not the case. Uniform convexity is really an isometric property of Banach spaces, which means that it depends on the norm and not only on the equivalence class of the norm.
For example, $\mathbb R^2$ with the Euclidean norm is uniformly convex, while $\mathbb R^2$ with the norm $\lVert \cdot\rVert_\infty$ is not uniformly convex, but of course these two norms are equivalent.