(Young’s inequality) Suppose a,b ∈ Rd are arbitrary vectors. Then, for any λ > 0 it holds that ⟨a, b⟩ ≤ (∥a∥^2/ 2λ )+((∥b∥^2) λ/ 2) . (Hint: use the Cauchy-Schwarz inequality ⟨a,b⟩ ≤ ∥a∥∥b∥.)

To prove Young's inequality for arbitrary vectors \( a, b \in \mathbb{R}^d \), we can use the Cauchy-Schwarz inequality as suggested in the hint.

### Step 1: Apply the Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states that for any vectors \( a \) and \( b \) in \( \mathbb{R}^d \):
\[
\langle a, b \rangle \leq \|a\|\|b\|
\]
This means that the dot product \( \langle a, b \rangle \) is less than or equal to the product of their magnitudes.

### Step 2: Use an Arbitrary Positive Scalar \( \lambda \)

Now, for any \( \lambda > 0 \), we can rewrite the product \( \|a\|\|b\| \) as:
\[
\|a\|\|b\| = \frac{\|a\|}{\sqrt{2\lambda}} \cdot \sqrt{2\lambda} \|b\|
\]

Applying the AM-GM (Arithmetic Mean-Geometric Mean) inequality on these terms, we get:
\[
\frac{\|a\|}{\sqrt{2\lambda}} \cdot \sqrt{2\lambda} \|b\| \leq \frac{1}{2}\left(\frac{\|a\|^2}{2\lambda} + 2\lambda \|b\|^2\right)
\]

This simplifies to:
\[
\|a\|\|b\| \leq \frac{\|a\|^2}{2\lambda} + \frac{\lambda \|b\|^2}{2}
\]

### Step 3: Relate to the Dot Product

Given that \( \langle a, b \rangle \leq \|a\|\|b\| \), we now conclude:
\[
\langle a, b \rangle \leq \frac{\|a\|^2}{2\lambda} + \frac{\lambda \|b\|^2}{2}
\]

### Conclusion

This completes the proof of Young's inequality:
\[
\langle a, b \rangle \leq \frac{\|a\|^2}{2\lambda} + \frac{\lambda \|b\|^2}{2}
\]
where \( \lambda > 0 \) is an arbitrary positive scalar.

Would you like further details on this proof, or do you have any other questions? Here are some related questions that might help you understand this topic better:

1. How does the AM-GM inequality apply in the context of Young's inequality?
2. Can Young's inequality be extended to more general norms or spaces?
3. What is the geometric interpretation of Young's inequality in \( \mathbb{R}^2 \) or \( \mathbb{R}^3 \)?
4. How is Young's inequality used in optimization problems?
5. How does Young's inequality relate to the concept of convexity?

**Tip:** Understanding inequalities like Young's and Cauchy-Schwarz is crucial, as they frequently appear in optimization and analysis problems.

Explore the proof of Young's inequality using Cauchy-Schwarz and AM-GM inequalities. Understand how this inequality applies to arbitrary vectors in Rd and its relevance in mathematical optimization and analysis. Suitable for advanced undergraduate students and researchers.

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