Find the Domain |p+q|^2+|p-q|^2=2|p|^2+2|q|^2

Problem

|p+q|2+|*p−q|2=2*|p|2+2|*q|2

Solution

1. Identify the given equation as the Parallelogram Law, which relates the lengths of the sides and diagonals of a parallelogram in the context of vectors or complex numbers.

2. Analyze the variables p and q In the context of this identity, p and q are typically elements of an inner product space, such as real numbers ℝ complex numbers ℂ or vectors in ℝn

3. Evaluate the constraints on the variables. The equation holds true for all possible values of p and q within these sets because it is a mathematical identity derived from the definition of the norm induced by an inner product.

4. Determine the domain based on the universal validity of the identity. Since there are no values that cause division by zero, square roots of negative numbers (in the real domain), or other undefined operations, the variables can be any value in their respective sets.

Final Answer

Domain: *p,q∈ℂ* (or any inner product space)

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