Question:
For \(\alpha,\beta\in V\), in an inner product space \(V(F)\), the triangle inequality states that
For \(\alpha,\beta\in V\), in an inner product space \(V(F)\), the triangle inequality states that
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Triangle inequality: norm of the sum is always less than or equal to the sum of norms.
Updated On: Jul 3, 2026
- \(\|\alpha+\beta\|=\|\alpha\|+\|\beta\|\)
- \(\|\alpha+\beta\|>\|\alpha\|+\|\beta\|\)
- \(\|\alpha+\beta\|<\|\alpha\|+\|\beta\|\)
- \(\|\alpha+\beta\|\leq \|\alpha\|+\|\beta\|\)
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The Correct Option is D
Solution and Explanation
Concept:
The triangle inequality is one of the fundamental properties of normed spaces. In an inner product space, the norm is defined by \[ \|\alpha\|=\sqrt{(\alpha,\alpha)} \] For any two vectors \(\alpha,\beta\in V\), \[ \|\alpha+\beta\|\leq \|\alpha\|+\|\beta\| \]
Step 1: Understand geometrical meaning. The expression \[ \|\alpha+\beta\| \] represents the length of the sum of two vectors. The triangle inequality says that the length of one side of a triangle cannot exceed the sum of the lengths of the other two sides.
Step 2: Write the standard result. \[ \|\alpha+\beta\|\leq \|\alpha\|+\|\beta\| \]
Step 3: Final answer. \[ \boxed{\|\alpha+\beta\|\leq \|\alpha\|+\|\beta\|} \]
The triangle inequality is one of the fundamental properties of normed spaces. In an inner product space, the norm is defined by \[ \|\alpha\|=\sqrt{(\alpha,\alpha)} \] For any two vectors \(\alpha,\beta\in V\), \[ \|\alpha+\beta\|\leq \|\alpha\|+\|\beta\| \]
Step 1: Understand geometrical meaning. The expression \[ \|\alpha+\beta\| \] represents the length of the sum of two vectors. The triangle inequality says that the length of one side of a triangle cannot exceed the sum of the lengths of the other two sides.
Step 2: Write the standard result. \[ \|\alpha+\beta\|\leq \|\alpha\|+\|\beta\| \]
Step 3: Final answer. \[ \boxed{\|\alpha+\beta\|\leq \|\alpha\|+\|\beta\|} \]
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