Abstract¶ The conditions for a functional of two vectors to be an inner product.
Keywords: Vectors Vector Spaces Inner Products. ¶ A vector space V V V inner product space .
An inner product, ( ⋅ , ⋅ ) (\cdot, \cdot) ( ⋅ , ⋅ ) V V V F n \mathbb{F}^{n} F n ( ⋅ , ⋅ ) : F n ⊗ F n → F 1 (\cdot, \cdot): \mathbb{F}^{n} \otimes \mathbb{F}^{n} \rightarrow \mathbb{F}^{1} ( ⋅ , ⋅ ) : F n ⊗ F n → F 1
Inner Product Conditions ¶ Let ψ , σ ∈ V \psi, ~\sigma ~\in~V ψ , σ ∈ V a ∈ F 1 a \in \mathbb{F}^{1} a ∈ F 1
To be an inner product the functional ( ⋅ , ⋅ ) (\cdot, \cdot) ( ⋅ , ⋅ )
( ψ , ψ ) ≥ 0 (\psi, \psi) \geq 0 ( ψ , ψ ) ≥ 0 ( ψ , ψ ) = 0 (\psi, \psi) = 0 ( ψ , ψ ) = 0 ψ = 0 \psi=0 ψ = 0
( ψ , ω ) = ( ω , ψ ) ∗ (\psi,\omega) = (\omega, \psi)^{*} ( ψ , ω ) = ( ω , ψ ) ∗
Conjugate symmetry
If V V V R \mathbb{R} R ( ψ , ω ) = ( ω , ψ ) (\psi,\omega) = (\omega, \psi) ( ψ , ω ) = ( ω , ψ )
( a ψ , ω ) = a ( ψ , ω ) (a\psi, \omega) = a (\psi, \omega) ( a ψ , ω ) = a ( ψ , ω )
Linearity in the first argument
If V V V C \mathbb{C} C ( a ψ , ω ) = a ∗ ( ψ , ω ) (a \psi,\omega) = a^*(\psi, \omega) ( a ψ , ω ) = a ∗ ( ψ , ω )
( ω + σ , ψ ) = ( ω , ψ ) + ( σ , ψ ) (\omega + \sigma, \psi) = (\omega, \psi) + (\sigma, \psi) ( ω + σ , ψ ) = ( ω , ψ ) + ( σ , ψ )
Implication ¶ Given a vector space V V V
( σ , ψ + ω ) = ( σ , ψ ) + ( σ , ω ) , ( ψ , a ω ) = a ∗ ( ψ , ω ) (\sigma, \psi + \omega) = (\sigma,\psi) + (\sigma, \omega), ~ ~ ~ (\psi, a\omega) = a^* (\psi, \omega) ( σ , ψ + ω ) = ( σ , ψ ) + ( σ , ω ) , ( ψ , aω ) = a ∗ ( ψ , ω ) which is called being anti-linear in the second argument. Hence, linearity in the first argument implies anti-linearity in the second argument.
( ψ , a ω ) = ( a ω , ψ ) ∗ = [ a ( ω , ψ ) ] ∗ = a ∗ ( ω , ψ ) ∗ = a ∗ ( ψ , ω ) \begin{align*}
(\psi, a\omega) &= (a \omega, \psi)^* \\
&= \big[a (\omega, \psi) \big]^* \\
&= a^* (\omega, \psi)^* \\
&= a^* (\psi, \omega)
\end{align*} ( ψ , aω ) = ( aω , ψ ) ∗ = [ a ( ω , ψ ) ] ∗ = a ∗ ( ω , ψ ) ∗ = a ∗ ( ψ , ω ) ( σ , ψ + ω ) = ( ψ + ω , σ ) ∗ = [ ( ψ , σ ) + ( ω , σ ) ] ∗ = ( ψ , σ ) ∗ + ( ω , σ ) ∗ = ( σ , ψ ) + ( σ , ω ) . \begin{align*}
(\sigma, \psi + \omega) &= (\psi + \omega, \sigma)^*\\
&= \big[(\psi, \sigma) + (\omega, \sigma) \big ]^* \\
&= (\psi, \sigma)^* + (\omega, \sigma)^* \\\
&= (\sigma, \psi) + (\sigma, \omega).
\end{align*} ( σ , ψ + ω ) = ( ψ + ω , σ ) ∗ = [ ( ψ , σ ) + ( ω , σ ) ] ∗ = ( ψ , σ ) ∗ + ( ω , σ ) ∗ = ( σ , ψ ) + ( σ , ω ) . Cauchy–Schwarz inequality ¶ The Cauchy–Schwarz inequality is an upper bound on the inner product between two vectors, in any vector space, in terms of the norms
Let x , y ∈ V \bm{x}, \bm{y}~\in~V x , y ∈ V V V V F \mathbb{F} F ( ⋅ , ⋅ ) → F 1 (\cdot, \cdot) \rightarrow \mathbb{F}^{1} ( ⋅ , ⋅ ) → F 1
∣ ( x , y ) ∣ 2 ≤ ∣ ∣ x ∣ ∣ ∣ ∣ y ∣ ∣ , \vert (\bm{x}, \bm{y}) \vert ^{2} \leq \vert \vert \bm{x} \vert \vert ~ \vert \vert \bm{y} \vert \vert, ∣ ( x , y ) ∣ 2 ≤ ∣∣ x ∣∣ ∣∣ y ∣∣ , where ∣ ⋅ ∣ \vert \cdot \vert ∣ ⋅ ∣
∣ ∣ x ∣ ∣ = ( x , x ) . \vert \vert \bm{x} \vert \vert = \sqrt{ (\bm{x}, \bm{x}) }. ∣∣ x ∣∣ = ( x , x ) .