Let \Lambda^* denote the dual lattice, let c > 0, prove (c\Lambda)* \subseteq (1/c)\Lambda^*.

To prove that \((c\Lambda)^* \subseteq \left(\frac{1}{c}\right) \Lambda^*\), we will use the definitions of the scaled lattice \(c\Lambda\) and the dual lattice \(\Lambda^*\). Let's break down the proof step by step.

### Definitions and Notations

1. **Lattice \(\Lambda\)**:
   \[
   \Lambda = \left\{ \sum_{i=1}^n a_i v_i \mid a_i \in \mathbb{Z} \right\}
   \]
   where \(v_1, v_2, \ldots, v_n\) are linearly independent vectors in \(\mathbb{R}^n\).

2. **Dual Lattice \(\Lambda^*\)**:
   \[
   \Lambda^* = \{ y \in \mathbb{R}^n \mid \forall v \in \Lambda, \langle y, v \rangle \in \mathbb{Z} \}.
   \]

3. **Scaled Lattice \(c\Lambda\)**:
   \[
   c\Lambda = \{ c v \mid v \in \Lambda \}.
   \]

4. **Dual of Scaled Lattice \((c\Lambda)^*\)**:
   \[
   (c\Lambda)^* = \{ y \in \mathbb{R}^n \mid \forall u \in c\Lambda, \langle y, u \rangle \in \mathbb{Z} \}.
   \]

### Proof

To show \((c\Lambda)^* \subseteq \left(\frac{1}{c}\right) \Lambda^*\), we need to show that any \(y \in (c\Lambda)^*\) is also in \(\left(\frac{1}{c}\right) \Lambda^*\).

1. **Condition for \(y \in (c\Lambda)^*\)**:
   By definition:
   \[
   y \in (c\Lambda)^* \implies \forall u \in c\Lambda, \langle y, u \rangle \in \mathbb{Z}.
   \]

2. **Expressing \(u \in c\Lambda\)**:
   Any vector \(u \in c\Lambda\) can be written as \(u = cv\) where \(v \in \Lambda\).

3. **Dot Product Condition**:
   For \(y \in (c\Lambda)^*\):
   \[
   \forall v \in \Lambda, \langle y, cv \rangle \in \mathbb{Z}.
   \]

4. **Simplifying the Condition**:
   \[
   \forall v \in \Lambda, \langle y, cv \rangle = c \langle y, v \rangle \in \mathbb{Z}.
   \]

5. **Implication**:
   This implies that:
   \[
   \forall v \in \Lambda, \langle y, v \rangle \in \frac{1}{c} \mathbb{Z}.
   \]
   Let \(\frac{1}{c} \mathbb{Z}\) denote the set of all rational numbers that can be expressed as \(\frac{k}{c}\) where \(k \in \mathbb{Z}\).

6. **Condition for Dual Lattice Scaling**:
   \[
   \langle y, v \rangle \in \frac{1}{c} \mathbb{Z} \quad \forall v \in \Lambda.
   \]
   This means that:
   \[
   \frac{1}{c} y \in \Lambda^* \implies y \in c \Lambda^*.
   \]

7. **Conclusion**:
   Since \(\frac{1}{c} y \in \Lambda^*\), we have:
   \[
   y \in \left(\frac{1}{c}\right) \Lambda^*.
   \]

Therefore, we have proven that:
   \[
   (c\Lambda)^* \subseteq \left(\frac{1}{c}\right) \Lambda^*.
   \]

Would you like any details or have any questions? Here are some questions you might find interesting:

1. How does the definition of a dual lattice extend to more complex lattice structures?
2. What are some applications of scaled lattices and their duals in engineering?
3. Can you explain the significance of dual lattices in number theory?
4. What role do lattices play in error-correcting codes?
5. How do dual lattices relate to the concept of Fourier transforms?

**Tip:** When dealing with lattice proofs, visualizing the geometric properties and relationships between vectors can greatly aid in understanding and constructing proofs.

Learn how to prove (cλ)* ⊆ (1/c)λ* using the definitions of dual lattices and scaled lattices. This advanced mathematical problem explores the relationships between lattice structures and their duals, suitable for advanced mathematics students.

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