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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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You have to look at infinite dimensional Banach spaces. For example, $X=\ell^2$, vector space of square-summable sequence of real numbers. Let $Y:=\{(x_n)_n, \exists k\in\Bbb N, x_n=0\mbox{ if }n\geq k\}$. It's a vector subspace of $X$, but not complete since it's not closed (it's in fact a strict and dense subset).

However, for two Banach spaces $X$ and $Y$, you can put norms on $X\times Y$ such that this space is a Banach space. For example, if $N$ is a norm on $\Bbb R^2$, define $\lVert(x,y)\rVert:=N(\lVert x\rVert_X,\lVert y\rVert_Y)$.

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    Thanks for the second half of your answer Davide. I realize that my question wasn't accurately phrased which caused some confusion, and I am already aware of the first half of your answer. But for the first part of your question, can the norm N on R^2 be any old norm, or does it have to have certain properties (in order to make (X x Y, N) Banach)?2012-05-28
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    All the norms on $\Bbb R^2$ are equivalent, so the choice doesn't matter.2012-05-28
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    dim(R^2) = 2 < infinity. Any two norms on a finite dimensional v. space are equivalent. I get you. Thanks2012-05-28
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    I found something on wiki: http://en.wikipedia.org/wiki/Direct_sum_of_modules#Direct_sum_of_Banach_spaces2012-05-28
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    I think X and Y have to be over the same field for this to work.2012-05-28