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Let $j:X \to \mathbb{R}$ be a continuous map, where $X$ is a reflexive Banach space which has a minimizer $u_0$: $$\min_{u \in X} j(u) = j(u_0).$$

Let $Y \subset X$, where $Y$ is a Banach space with its own norm. The embedding is continuous. Does it follow that there exists a $y_0 \in Y$ such that $$\min_{y \in Y}j(y) = j(y_0)$$ i.e., does $j$ have a minimizer in $Y$ too?

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In general it is not true. Consider, for example, $X = \mathbb{R}^2$, $Y = \{0\}\times\mathbb{R}$, $$ j(x_1, x_2) = (x_1-1)^2 e^{-x_2^2}, \qquad (x_1, x_2)\in X. $$ Clearly, all points $(1, x_2)$ are minimizers of $j$ in $X$. On the other hand, $j(0, x_2) = e^{-x_2^2}$, hence $j$ does not have minimizers in $Y$.

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