Abstract
sents a real circle with a positive radius having the origin in its interior (because the constant term in this equation is negative); and when c, d are not both zero, the straight line represented by the equation (4) meets this circle in two real and distinct points. We can, therefore, always find (at least) two distinct complex numbers zs such that z =zk satisfy the equation (1). This proves slightly more than what we set out to prove. For real inner product spaces X we may similarly reduce the proof of the corresponding theorem to showing that a certain quadratic equation with real coefficients has real roots. The author wishes to thank Professor V. Ganapathy Iyer for his encouragement and guidance.
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