تطبيقات وامثلة عن فضاء الضرب الداخلي

An inner product space is a vector space V along with a function h, i called an inner product which
associates each pair of vectors u, v with a scalar hu, vi, and which satisfies:
(1) hu, ui  0 with equality if and only if u = 0
(2) hu, vi = hv, ui and
(3) hu + v,wi = hu,wi + hv,wi
Combining (2) and (3), we also have hu, v +wi = hu, vi+hu,wi. Condition (1) is called positive definite,
condition (2) is called symmetric and condition (3) with the note above is called bilinear. Thus an inner
product is an example of a positive definite, symmetric bilinear function or form on the vector space V .
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