If a sequence of functions $u_k$ converges to $u$ in a Banach space $X$, and $X$ is continuously embedded in the Banach space $Y$, then does $u_k$ converges to same $u$ in $Y$?
About continuous embedding
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functional-analysis
1 Answers
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Yes. Let $\iota\colon X \to Y$ denote the embedding. If $u_n \to u$ in $X$, we have by continuity of $\iota$ that $u_n = \iota u_n \to \iota u = u$ in $Y$.