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 ${\displaystyle {\mathbb{ℝ}}^n}$ endowed with the real inner product ${\displaystyle ⟨x,y⟩={\displaystyle \sum _{k=1}^n}{x}_k{y}_k}$ for all ${\displaystyle x=(x_1,\dots ,x_n),y=(y_1,\dots ,y_n)\in {\mathbb{ℝ}}^n}$. This inner product induces the Euclidean norm ${\displaystyle \Vert x\Vert =⟨x,x{⟩}^{1/2}}$
2. The space ${\displaystyle L^2(\mathbb{ℝ})}$ of the equivalence classes of all complex-valued Lebesgue measurable scalar square integrable functions on ${\displaystyle \mathbb{ℝ}}$ with the complex inner product ${\displaystyle ⟨f,g⟩={\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}}f(x)\overline{g(x)}dx}$. Here a square integrable function is any function f satisfying ${\displaystyle {\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}}|f(x){|}^{2}dx<\mathrm{\infty }}$. The inner product induces the norm ${\displaystyle \Vert f\Vert ={({\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}}|f(x){|}^{2}dx)}^{1/2}}$