# Distance between real finite dimensional linear subspaces

Is there a usual distance between linear subspaces ($V,W$) of an n-dimensional normed vector space with inner product?

In the case of hyper-planes one could use the angle (based on the inner product of the vector space). What can be used in the case of subspaces with lower dimension (not necessarily equal)? e.g. $dim(V)= n-2$ and $dim(W) = n-4$

Thanks

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Well, the word "distance" is already taken. If you call something "distance", you want it to be a distance, i.e. a metric. Your question is rather vague, since you do not add any reasonable condition that your "distance" should satisfy. – Siminore Sep 17 '12 at 17:03
I need the usual properties of a distance (maybe measure?) d(x,x)=0 iff x=x, d(x,y)>= 0 , etc... – JuanPi Sep 17 '12 at 17:49

There is, if both subspaces have the same dimension. You can actually make the set of $k$-dimensional subspaces of a vector space $V$ into a metric space (a manifold, in fact) called the Grassmannian, denoted $\mathrm{Gr}(k,V)$. The distance between two subspaces $W$ and $W'$ is then $\|P_W-P_{W'}\|$ where $P_X$ denotes projection onto $X$ and $\|\cdot\|$ is the operator norm.
@JuanPi Then there is no notion I know of. If the dimension of $V$ is $n$ and the dimensions of $W,W'$ are $k,n-k$ we can use the duality of $k$ and $n-k$ dimensional subspaces to consider both as $k$ dimensional subspaces, but otherwise I have no idea. – Alex Becker Sep 17 '12 at 17:55