An inner product is a function \( \left\langle \cdot , \cdot \right\rangle : V \times V \to F \) where \( V \) is a vector space over the field \( F \), it assigns to each pair \(\mathbf{v}, \mathbf{w} \in V\) a real number \(\langle\mathbf{v}, \mathbf{w}\rangle\) such that, for all \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in V\) and \(\alpha \in \mathbf{F}\) and satisfies:
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\(\langle\mathbf{v}, \mathbf{v}\rangle \geq 0\), with equality if and only if \(\mathbf{v}=\mathbf{0}\).
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\(\langle\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{w}, \mathbf{v}\rangle\).
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\(\langle\mathbf{u}+\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{w}\rangle+\langle\mathbf{v}, \mathbf{w}\rangle\),
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\(\langle\alpha \mathbf{v}, \mathbf{w}\rangle=\alpha\langle\mathbf{v}, \mathbf{w}\rangle\).