Let $V$ be an inner product space of finite dimension $n$ over its field of scalars. Show that there exists a subset $\{v_1,...,v_{2n}\}$ of $V$ satisfying the conditions that $\langle v_i, v_j \rangle \leq 0$ for all $1 \leq i \neq j \leq 2n$.

I know that if the inner product is equal to zero than the two vectors are orthogonal. Also, correct me if I'm wrong, but for the inner product to be $<0$ that would imply that the angle between them is greater than 90 degrees.

That's about as far as I'm getting here. Is there a need for the Gram-Schmidt Process?

  • 1
    $\begingroup$ In $\Bbb R^3$, think of an octahedron. $\endgroup$
    – pjs36
    Commented Nov 28, 2015 at 22:25
  • $\begingroup$ @I.Cavey What is the inner product of $(1,1)$ and $(-1,-1)$? $\endgroup$ Commented Nov 28, 2015 at 22:34
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    $\begingroup$ @DavidC.Ullrich Ah, thank you for letting me realize my own mistake. $\endgroup$
    – Ian Cavey
    Commented Nov 28, 2015 at 22:38

1 Answer 1


Pick an orthonormal basis for your space, call it $B = \{e_1, e_2, \ldots, e_n\}$. Such a thing exists, of course, thanks to Gram-Schmidt process.

Now for each $e_i$, let $f_i = -e_i$. You can verify that $\langle e_i, f_i \rangle = -1$, while all $e_i$ are orthogonal to all $f_j$ when $j \neq i$. Of course the various $f_i$ are related to each other in the same way the various $e_i$ are.

In $\Bbb R^n$, the convex hulls of these $2n$ vectors are the $n$-dimensional cross-polytopes, dual to well-known $n$-dimensional cubes.


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