A Banach space which is not an inner product space.

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The supremum norm on C[a, b] can not be obtained from an inner product. Since if f(t) = 1 and g(t) = \frac{t−a}{b−a}, then ∥ f ∥=∥ g ∥= 1, ∥ f − g ∥= sup\{|1 − \frac{t−a}{b−a}|; t ∈ [a, b]\} = 1 and ∥ f + g ∥= sup\{|1 + \frac{t−a}{b−a}|; t ∈ [a, b]\} = 2 and so the parallelogram equality ∥ f + g ∥^{2} + ∥ f − g ∥^{2}= 2 ∥ f ∥^{2} +2 ∥ g ∥^{2} (which is satisfied in every inner product space) isn’t held.

Comment. Indeed this Banach space is not an inner product space in any equivalent norm.