Inner product space

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In mathematics, an inner product space is a vector space that is endowed with an inner product. It is also a normed space since an inner product induces a norm on the vector space on which it is defined. A complete inner product space is called a Hilbert space.

## Examples of inner product spaces[edit]

1. The Euclidean space ℝn endowed with the real inner product ⟨x,y⟩=∑k=1nxkyk for all x=(x1,…,xn),y=(y1,…,yn)∈ℝn. This inner product induces the Euclidean norm ‖x‖=⟨x,x⟩1/2
2. The space L2(ℝ) of the equivalence classes of all complex-valued Lebesgue measurable scalar square integrable functions on ℝ with the complex inner product ⟨f,g⟩=∫−∞∞f(x)g(x)‾dx. Here a square integrable function is any function f satisfying ∫−∞∞|f(x)|2dx<∞. The inner product induces the norm ‖f‖=(∫−∞∞|f(x)|2dx)1/2