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If $A$ is an $n\\times n$ matrix with $|\\sum_{i,j=1}^na_{i j} s_i t_j | \\leq 1$ for all vectors $s, t$ with $|s_i|, |t_i| \\leq 1,$ then there exists a constant $K_G(n),$ independent of the choice of $A,$ such that $$\\sup |\\sum_{i,j}^n a_{i j} <x_i , y_j > | = K_G(n),$$ where the supremum is taken over all unit vectors $x_1,..., x_n; y_1,...,y_n$ of a Hilbert space $H.$ Grothendieck proved that the limit of $K_G(n)$ remains finite as $n \\to \\infty$ and is the universal constant $K_G$ known as the Grothendieck constant. The exact value of the constant $K_G,$ which depends no the ground field, is not known. We will give an elementary proof, due to Kirvine, of this inequality and discuss many of its surprising consequences.