Let X be an inner product space with the inner product given by <, >. For x = X, define the function ||· ||: X → K given by ||x|| = < x, − x >1/2, the non negative square root of < x, x>. Show that ||- ||: X→ K defines a norm on X and |< (x, y) >| ≤ ||x|| ||y|| for all x, y = X. Also show that for all x, ye X, |||x + y||² + ||x − y||²³ = 2 (|×||² + 1y1²).

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