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For questions about inner products and inner product spaces, including questions about the dot product. An inner product space is a vector space equipped with an inner product. The dot product (seen in multivariable calculus and linear algebra) is a simple example of an inner product—other inner products may be seen as generalizations of the dot product.

Given vectors $$x = (x_1, x_2, \dotsc, x_n)$$ and $$y = (y_1, y_2, \dotsc, y_n)$$ in $$\mathbb{R}^n$$, the dot product of $$x$$ and $$y$$ is $$x \cdot y = \sum_{j=1}^{n} x_j y_j.$$ The dot product on $$\mathbb{R}^n$$ is linear in both $$x$$ and $$y$$ and has the property that $$x\cdot x \ge 0$$ for all $$x$$, with equality if and only if $$x = 0$$. Moreover $$x \cdot y = \lVert x\rVert \lVert y\rVert \cos(\theta)$$, where $$\lVert x\rVert$$ denotes the length of $$x$$ and $$\theta$$ is the measure of the angle between the vectors $$x$$ and $$y$$. The dot product is then an algebraic tool which can be used to describe geometric properties of $$\mathbb{R}^n$$ (e.g. distance and angle).

An inner product is a generalization of the dot product. An inner product space is a vector space over a field $$\mathbb K$$ (either $$\mathbb R$$ or $$\mathbb C$$) endowed with a map $$\langle\cdot,\cdot\rangle\colon V\times V\longrightarrow\mathbb K$$ such that

1. $$(\forall v_1,v_2,v\in V):\langle v_1+v_2,v\rangle=\langle v_1,v\rangle+\langle v_2,v\rangle$$;
2. $$(\forall v_1,v_2\in V)(\forall\lambda\in\mathbb{K}):\langle\lambda v_1,v_2\rangle=\lambda\langle v_1,v_2\rangle$$;
3. $$(\forall v_1,v_2\in V):\langle v_1,v_2\rangle=\overline{\langle v_2,v_1\rangle}$$;
4. $$(\forall v\in V):\langle v,v\rangle\geqslant0$$ and $$\langle v,v\rangle=0\iff v=0$$.

Such a map is called an inner product. As an example, consider the space $$\mathcal{C}\bigl([0,1]\bigr)$$ of all continuous functions from $$[0,1]$$ into $$\mathbb C$$. If $$f,g\in\mathcal{C}\bigl([0,1]\bigr)$$, define$$\langle f,g\rangle=\int_0^1f(t)\overline{g(t)}\ \mathrm dt.$$