To show that the function $f$ defined by

$f\left((x_1, x_2), (y_1, y_2)\right) = |x_1 y_1| + |x_2 + y_2|$

is not an inner product on $\mathbb{R}^2$, we need to check if it satisfies the properties of an inner product. The properties that a function $\langle \cdot, \cdot \rangle: \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}$ must satisfy to be an inner product are:

1. Positivity: $\langle \mathbf{x}, \mathbf{x} \rangle \geq 0$ and $\langle \mathbf{x}, \mathbf{x} \rangle = 0$ if and only if $\mathbf{x} = \mathbf{0}$.
2. Linearity in the first argument: $\langle a\mathbf{x} + b\mathbf{y}, \mathbf{z} \rangle = a\langle \mathbf{x}, \mathbf{z} \rangle + b\langle \mathbf{y}, \mathbf{z} \rangle$ for all $a, b \in \mathbb{R}$ and all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^2$.
3. Symmetry: $\langle \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{y}, \mathbf{x} \rangle$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^2$.

Let's verify these properties one by one for the function $f$.

1. Positivity

Consider $f\left((x_1, x_2), (x_1, x_2)\right)$: $f\left((x_1, x_2), (x_1, x_2)\right) = |x_1 \cdot x_1| + |x_2 + x_2| = |x_1^2| + |2x_2|$ Clearly, $f\left((x_1, x_2), (x_1, x_2)\right) \geq 0$. However, it equals 0 only if $x_1 = 0$ and $x_2 = 0$. So, the positivity condition is satisfied.

2. Linearity in the First Argument

Consider the function for $a, b \in \mathbb{R}$ and vectors $(x_1, x_2), (y_1, y_2), (z_1, z_2) \in \mathbb{R}^2$: $f\left(a(x_1, x_2) + b(y_1, y_2), (z_1, z_2)\right) = f\left((ax_1 + by_1, ax_2 + by_2), (z_1, z_2)\right)$ This gives: $f\left((ax_1 + by_1, ax_2 + by_2), (z_1, z_2)\right) = |(ax_1 + by_1)z_1| + |(ax_2 + by_2) + z_2|$ However: $a f\left((x_1, x_2), (z_1, z_2)\right) + b f\left((y_1, y_2), (z_1, z_2)\right) = a\left(|x_1 z_1| + |x_2 + z_2|\right) + b\left(|y_1 z_1| + |y_2 + z_2|\right)$ Linearity fails because $|(ax_1 + by_1)z_1|$ is generally not equal to $a|x_1 z_1| + b|y_1 z_1|$, especially if $x_1 y_1 z_1 < 0$.

3. Symmetry

Let's check if $f\left((x_1, x_2), (y_1, y_2)\right) = f\left((y_1, y_2), (x_1, x_2)\right)$: $f\left((x_1, x_2), (y_1, y_2)\right) = |x_1 y_1| + |x_2 + y_2|$ $f\left((y_1, y_2), (x_1, x_2)\right) = |y_1 x_1| + |y_2 + x_2|$ Since $|x_1 y_1| = |y_1 x_1|$ and $|x_2 + y_2| = |y_2 + x_2|$, symmetry is satisfied.

Conclusion

The function fails to satisfy the linearity condition, which is a critical requirement for an inner product. Therefore, the function $f\left((x_1, x_2), (y_1, y_2)\right) = |x_1 y_1| + |x_2 + y_2|$ is not an inner product on $\mathbb{R}^2$.

Would you like to explore any particular property further, or do you have any other questions?

Here are five related questions you might find interesting:

1. What are the properties that distinguish an inner product from a general bilinear form?
2. How does the Cauchy-Schwarz inequality relate to inner products?
3. Can you provide examples of valid inner products on $\mathbb{R}^n$?
4. How do inner products define the notion of orthogonality in vector spaces?
5. What are some applications of inner product spaces in functional analysis?

Tip: When verifying if a function is an inner product, always start by checking linearity, as it is often the most likely condition to fail.