Get the free "Surface plot of f(x, y)" widget for your website, blog, Wordpress, Blogger, or iGoogle Find more Engineering widgets in WolframAlphaHx,yi2 ≤ hx,xihy,yi for all x,y ∈ V Proof For any t ∈ R let vt = xty Then hvt,vti = hx,xi2thx,yit2hy,yi The righthand side is a quadratic polynomial in t (provided that y 6= 0) Since hvt,vti ≥ 0 for all t, the discriminant D is nonpositive But D = 4hx,yi2 −4hx,xihy,yi CauchySchwarz Inequality hx,yi ≤ p hx,xi p hy,yi CauchySchwarz Inequality hx,yi ≤ p hx,xi pR s E { s E s ŃC v g ÂȂ畟 R s Ò i i j g Ȉ @ C Y C v g I t B X j ߌ ͏j ̂ T ͐f ÁB E i w k T E R C ^ ԂŖ R Pdf A Strengthening Of Erdos Gallai Theorem And Proof Of Woodall S Conjecture "Y "¯^ p[} x[V[g