Inner Product

Let ${\mathbb V}$ be a linear vector space over the field ${\mathbb F}={\mathbb R} \textrm{ or }{\mathbb C}$. Let $v,\ w$ denote vectors in ${\mathbb V}$ — the same vectors will be written as the ket vectors $|v\rangle,\ |w\rangle$ in Dirac notation. An inner product is a map from ${\mathbb V}\times {\mathbb V}$ to the field ${\mathbb F}$. We write

\begin{eqnarray} {\mathbb V}\times {\mathbb V} &\rightarrow& {\mathbb F} \\ (v,w) &\mapsto & \langle v, w \rangle \\ (|v\rangle,|w\rangle) &\mapsto & \langle v| w \rangle \ \textrm{in Dirac notation}\ . \end{eqnarray}

The inner product satisfies the following conditions

  • $\langle v , w \rangle^* = \langle w, v \rangle$. So for real linear vector spaces, the inner product is symmetric.
  • It is linear in the second entry $\langle v, a w_1+b w_2 \rangle = a \langle v , w_1 \rangle + b \langle v , w_2\rangle$ for $a,b \in {\mathbb F}$. Using the first condition, we see that it is anti-linear in the first entry. Caution: We are following the physics convention here — in the standard maths convention the roles of the first and second entries are exchanged. This enables us to relate to the Dirac notation by replacing the comma by a $|$.
  • $\langle v, v \rangle=0$ implies that $v=0$. This implies that the only vector that has zero-norm is the zero-vector.

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