Step 1: Understanding the Concept:
This question asks for the statement of the triangle inequality for the norm induced by an inner product. The norm of a vector \(\alpha\), denoted \(||\alpha||\), is defined as \(\sqrt{\langle \alpha, \alpha \rangle}\). The triangle inequality is a fundamental property of any norm.
Step 2: Key Formula or Approach:
The triangle inequality is a theorem derived from the axioms of an inner product space. It relates the norm of a sum of two vectors to the sum of their individual norms.
The theorem states: For any two vectors \(\alpha, \beta\) in an inner product space,
\[ ||\alpha+\beta|| \le ||\alpha|| + ||\beta|| \]
Equality holds if and only if one vector is a non-negative scalar multiple of the other.
Step 3: Detailed Explanation:
The statement of the triangle inequality is a direct definition/theorem. Let's analyze the options:
- (A) \(||\alpha+\beta|| = ||\alpha|| + ||\beta||\): This is the case of equality, which only holds under specific conditions (when the vectors are in the same direction). It is not the general inequality.
- (B) \(||\alpha+\beta|| > ||\alpha|| + ||\beta||\): This is never true and contradicts the inequality.
- (C) \(||\alpha+\beta|| < ||\alpha|| + ||\beta||\): This is the strict inequality, which holds when the vectors are not linearly dependent in the same direction. The general theorem includes the possibility of equality.
- (D) \(||\alpha+\beta|| \le ||\alpha|| + ||\beta||\): This is the correct and complete statement of the triangle inequality. It accounts for both the strict inequality and the case of equality.
The name "triangle inequality" comes from the geometric interpretation in \(\mathbb{R}^2\) or \(\mathbb{R}^3\): the length of any side of a triangle is less than or equal to the sum of the lengths of the other two sides. If we consider the vectors \(\alpha\), \(\beta\), and \(\alpha+\beta\) forming a triangle, their lengths (norms) must satisfy this property.
Step 4: Final Answer:
The triangle inequality is stated as \(||\alpha+\beta|| \le ||\alpha|| + ||\beta||\), which corresponds to option (D).