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Suppose $(Y,\Vert\cdot\Vert)$ is a complete normed linear space. If the vector space $X\supset Y$ with the same norm $\Vert\cdot\Vert$ is a normed linear space, then is $(X,\Vert\cdot\Vert)$ necessarily complete?

My guess is no. However, I am not aware of any examples.

Side interest: If X and Y are Banach (with possibly different norms), I want to make $X \times Y$ Banach. But I realize that in order to do this, we cannot use the same norm as we did for $X$ and $Y$ because it's not like $X \subseteq X \times Y$ or $Y \subseteq X \times Y$. What norm (if there is one) on $X \times Y$ will garuntee us a Banach space?

I'm sure these questions are standard ones in functional analysis. I just haven't come across them in my module. Thanks in advance.

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    Should $X\supset Y$ be $X\subset Y$? Otherwise "the same norm" doesn't make sense. In the $\subset$ case, it would be the restricted norm (which is strictly speaking not the same).2012-05-28
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    Just take $X$ to be any incomplete space (necessarily infinite dimensional) and $Y$ any finite dimensional subspace.2012-05-28
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    Yes you are right as well.2012-05-28

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