Answer to Question #267036 in Functional Analysis for Gestavo

Question

Proof whether the following operations are inner product operations:

⟨x, y⟩ = 2x₁y₁ − x₁y₂ − x₂y₁ + 2x₂y₂, x=(x₁, x₂), y=(y₁, y₂)

Answer 1

We must verify linearity,symmetry and positive definiteness properties.

We can define <x,y> through matrix operations as

$<x,y>=x^T\cdot A\cdot y, \space x=\begin{pmatrix} x_1 \\ x_2\end{pmatrix},x^T=(x_1\space x_2),y=\begin{pmatrix} y_1 \\ y_2\end{pmatrix}, A=\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$

and use linearity of transposition operation $^T$ and matrix multiplication

So we have

$<x^{(1)}+x^{(2)},y>=(x^{(1)}+x^{(2)})^T\cdot A\cdot y=\\ ( (x^{(1)})^T+(x^{(2)}))\cdot A\cdot y=((x^{(1)})^T\cdot A+(x^{(2)})^T\cdot A\cdot y=\\ (x^{(1)})^T\cdot A\cdot y+(x^{(2)})^T\cdot A\cdot y=<x^{(1)},y>+<x^{(2)},y>$

and

$<c\cdot x,y>=(c\cdot x)^T\cdot A \cdot y=(c\cdot x^T)\cdot A\cdot y=\\ c\cdot (x^T\cdot A\cdot y)=c\cdot <x,y>$

Thus linearity by the first argument of <,> is proved.

2) Symmetry property

$<x,y>=x^T\cdot A \cdot y=[x^T\cdot A \cdot y\in R]=(x^T\cdot A \cdot y)^T=\\ y^T\cdot A^T\cdot (x^T)^T=[A^T=A,\space A-symmetric,(x^T)^T=x]=y^T\cdot A\cdot x=<y,x>$

From symmetry of <x,y> linearity by 2-argument is proved also.

3) Positive definiteness property.

This property better to verify by using simplest definition of <x,y>.

Let $x\ne\overline 0$ or $x_1^2+x_2^2> 0$ . We have

<x,x>=$2x_1^2-2x_1x_2+2x_2^2=(x_1-x_2)^2+x_1^2+x_2^2>0$

Thus all properties of inner product is proved.