Show that function taking an ordered pair (x_1, x_2), (y_1, y_2) of elements of R^2 to |x_1 y_1| + |x_2 + y_2| in not an inner product on R^2

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.

Show that function taking an ordered pair (x_1, x_2), (y_1, y_2) of elements of R^2 to |x_1 y_1| + |x_2 + y_2| is not an inner product on R^2

Explore a detailed proof showing that the function f((x_1, x_2), (y_1, y_2)) = |x_1 y_1| + |x_2 + y_2| is not an inner product on R^2. This explanation covers key concepts in linear algebra, including inner product properties such as positivity, linearity, and symmetry. Suitable for undergraduate students.

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