Question

Let \(X\) be a normed linear space. Prove that for any \(x, y \in X\) we have \(|\|x\|-\|y\|| \leq\|x-y\|\) Hint: Triangle inequality, \(\|x\|=\|(x-y)+y\|\).

Short answer

The inequality \[ |orm{x} - orm{y}| \leq orm{x - y} \] can be proved using the triangle inequality applied to both \(x\) and \(y\).

Step-by-step solution

Understand the Problem

We need to prove the inequality \[ |orm{x} - orm{y}| \leq orm{x-y} \] for any vectors \(x\) and \(y\) in a normed linear space \(X\).

Use Hint and Initial Setup

The hint given is to utilize the triangle inequality and the expression \[ orm{x} = orm{(x - y) + y} \] Start by rewriting \(x\) as \((x-y) + y\).

Apply the Triangle Inequality

Using the expression \[ orm{x} = orm{(x - y) + y} \], apply the triangle inequality: \[ orm{x} \leq orm{x - y} + orm{y} \] This gives us one side of the inequality we need.

Rearrange Using Symmetry

Now consider the expression \[ orm{y} = orm{(y - x) + x} \]. Using the triangle inequality again, we get: \[ orm{y} \leq orm{y - x} + orm{x} \] which is the same as \[ orm{y} \leq orm{x - y} + orm{x} \].

Combine Inequalities

The two expressions we have are: \[ orm{x} \leq orm{x - y} + orm{y} \] \[ orm{y} \leq orm{x - y} + orm{x} \] Subtract \(orm{y}\) from both sides of the first expression to get: \[ orm{x} - orm{y} \leq orm{x - y} \] Subtract \(orm{x}\) from both sides of the second expression to get: \[ orm{y} - orm{x} \leq orm{x - y} \]

Conclude the Proof

Recall that \(|a - b| = \text{max}(a - b, b - a)\). Thus, combining the inequalities, we get: \[ |orm{x} - orm{y}| \leq orm{x - y} \] This completes the proof.

Key concepts

Triangle Inequality

The triangle inequality is a fundamental concept in mathematics and it plays a crucial role in many proofs, including the one we are looking at. The triangle inequality states that for any vectors \(a\) and \(b\) in a normed space, the inequality \(|a + b| \leq |a| + |b|\) always holds. This means that the length (or norm) of the sum of two vectors is always less than or equal to the sum of their lengths.
In our proof, we use this principle twice. First, we apply it to the expression \(|x| = |(x - y) + y|\) to show that \(|x| \leq |x - y| + |y|\).
We then use it again in a symmetric form by looking at \(|y| = |(y - x) + x|\) to conclude that \(|y| \leq |y - x| + |x|\), which simplifies as \(|y| \leq |x - y| + |x|\).
These applications of the triangle inequality help us bound the norms of the vectors and drive toward the final inequality we need to prove.

Normed Linear Space

A normed linear space, also known as a normed vector space, is a vector space equipped with a function called a norm. This norm assigns a non-negative length or size to each vector in the space. Formally, a norm is a function \(|| \cdot ||\) that satisfies three properties:

• Norm positivity: \(||x|| \geq 0\) for all vectors \(x\), and \(||x|| = 0\) if and only if \(x = 0\).
• Scalar multiplication: \(||\alpha x|| = |\alpha| \cdot ||x||\) for any scalar \(\theta\) and vector \(x\).
• Triangle inequality: \(||x + y|| \leq ||x|| + ||y||\) for all vectors \(x\) and \(y\).

Understanding these properties helps in comprehending how vectors behave under operations and applying these properties, as seen in our proof, provides the structure needed to bound norms and show inequalities between the lengths of vectors.

Vector Norms

A vector norm is a measure of the 'size' or 'length' of a vector in a normed space. It is a function that takes a vector as input and returns a non-negative scalar. Common types of norms include:

• Euclidean norm: \(||x||_2 = \sqrt{x_1^2 + x_2^2 + ... + x_n^2}\), which is the most familiar and measures the 'straight-line' distance in Euclidean space.
• Manhattan norm: \(||x||_1 = |x_1| + |x_2| + ... + |x_n|\), which measures the distance traveled along grid lines.
• Maximum norm: \(||x||_\infty = \max(|x_1|, |x_2|, ..., |x_n|)\).

In the context of our proof, we assume a general norm without specifying a particular type. The essential properties of the norm, especially the triangle inequality, allow us to derive meaningful relationships between vector lengths.

Proof Techniques

Proving inequalities in normed linear spaces often involves a combination of algebraic manipulation and the application of fundamental properties like the triangle inequality. Here are some key techniques used in the proof:

• Rewriting Expressions: We start by rewriting the vector \(x\) as \( (x - y) + y \) to set up a scenario where the triangle inequality can be applied.
• Applying the Triangle Inequality: We use the fundamental property \( ||x + y|| \leq ||x|| + ||y|| \) to bound the norms of the vectors.
• Symmetry and Rearrangement: Recognizing symmetry in norms (e.g., \( ||x - y|| = ||y - x|| \) ) allows us to apply the triangle inequality to both \(x\) and \(y\).
• Combining Results: We consolidate inequalities from different steps to derive the desired result, using the definition of absolute value \( |a - b| = \text{max}(a - b, b - a) \) to show the final inequality.

The combination of these methods constructs a logical flow from the initial setup to the final proof, demonstrating the inequality we set out to prove.