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I'm trying to follow the The Farkas-Minkowski Theorem but I'm having a little bit of difficulty.

On the second page the author states,

Then we consider a set of the form $R_k := \left\{ z \mid z= \sum_{a}^{n} \mu_j a_j , \mu_j >= 0 \right\}$

First, suppose that $R_k$ contains the vectors $-a_1, -a_2, ..., -a_k$. Then $R_k$ is a subspace of dimension not exceeding k so it is closed.

I don't understand how they make the conclusion that $R_k$ is a dimension not exceeding so it is closed. I don't understand why it doesn't exceed the dimension or what that means and what that means in terms of why its closed or not.

That one line is very confusing for me and I must be missing something.

Thanks

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1 Answer 1

up vote 1 down vote accepted

Every subspace of a finite dimensional space is closed. There are many ways to see this,

I would go like this: In finite dimensional space, every linear subspace is of the form $\{x; Bx=0\}$ for some matrix B, i.e. it's a kernel of some linear function. Every linear function on $\mathbb R^n$ is continuous.

However, you can find other ways to prove it. Quick google search for finite dimensional subspace closed returns e.g. http://planetmath.org/encyclopedia/EveryFiniteDimensionalSubspaceOfANormedSpaceIsClosed.html

http://blogs.ethz.ch/kowalski/2008/10/23/finite-dimensional-normed-vector-spaces/

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Ok, thanks that makes sense –  coderdave May 1 '11 at 9:08

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