Question

In the text, the modulus is defined in terms of the inner product. Prove that this can be turned around by showing that for every ayer",

(x,y)=(|x + y| ² -|x-y|² )/ 4

Answer 1

To prove that for every pair (x,y), (x,y)=(|x + y| ² -|x-y|² )/ 4, we can use the definition of modulus in terms of the inner product.

Step-by-step explanation:

To prove that for every pair (x,y), (x,y)=(|x + y| ² -|x-y|² )/ 4, we can use the definition of modulus in terms of the inner product. The modulus of a vector x is defined as |x| = √(x·x), where · represents the inner product.

So, let's start with the left side of the equation. If we expand it, we get (x,y) = x·y, which is the inner product of x and y.

Now, let's consider the right side of the equation. Expanding it, we have (|x + y| ² -|x-y|² )/ 4 = (x+y)·(x+y)/4 - (x-y)·(x-y)/4 = (x·x + 2x·y + y·y - x·x + 2x·y - y·y)/4 = (4x·y)/4 = x·y.

Since both sides of the equation are equal to x·y, we have proved the given statement.