The goal is to prove that \( \mathbb{E}[L^2 f(x)] = \mathbb{E}[L f(x)] \), where \( L \) is a linear operator and \( \mathbb{E} \) represents the expectation.

### Assumptions
1. The linear operator \( L \) acts on the function \( f(x) \), where \( f(x) \) is some function.
2. The expectation operator \( \mathbb{E} \) is linear.

### Proof Outline

1. **Linearity of the Operator \( L \):**  
   A linear operator \( L \) satisfies:
   \[
   L(a f(x) + b g(x)) = a L(f(x)) + b L(g(x))
   \]
   for any functions \( f(x), g(x) \) and scalars \( a, b \). 

2. **Applying the Operator Twice:**
   Since we are dealing with \( L^2 \), we are applying the operator \( L \) twice:
   \[
   L^2 f(x) = L(L(f(x))).
   \]

3. **Expectation of \( L^2 f(x) \):**  
   Now consider the expectation:
   \[
   \mathbb{E}[L^2 f(x)] = \mathbb{E}[L(L(f(x)))].
   \]

4. **Commuting the Expectation and Linear Operator:**  
   If \( L \) and \( \mathbb{E} \) commute, i.e., if the linear operator \( L \) can be taken outside the expectation (this is typically true for many linear operators under regularity conditions), then we can write:
   \[
   \mathbb{E}[L(L(f(x)))] = L(\mathbb{E}[L(f(x))]).
   \]
   
5. **Final Step:**  
   Since \( \mathbb{E}[L(f(x))] = L(\mathbb{E}[f(x)]) \), we can write:
   \[
   \mathbb{E}[L^2 f(x)] = L(\mathbb{E}[L(f(x))]) = \mathbb{E}[L f(x)].
   \]

Thus, we have shown that:
\[
\mathbb{E}[L^2 f(x)] = \mathbb{E}[L f(x)].
\]

### Assumptions:
- This proof assumes that \( L \) and \( \mathbb{E} \) commute and that \( L \) is a linear operator.

Would you like more details or have any specific questions?

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Here are 5 questions to expand on this concept:
1. How does linearity of the expectation operator play a role in this proof?
2. Under what conditions do the linear operator \( L \) and the expectation \( \mathbb{E} \) commute?
3. Can you provide an example where \( L \) is a differential operator? How would this affect the proof?
4. What changes in the proof if \( L \) is not linear?
5. How would this result differ if the expectation operator were taken after applying \( L^2 f(x) \)?

**Tip:** When working with linear operators, understanding how they interact with integrals or sums (like expectations) is crucial to simplifying complex expressions.

Explore the proof of E[L^2 f(x)] = E[L f(x)] by utilizing the linearity of expectation and linear operators. This advanced topic in functional analysis is suitable for upper-level undergraduates or graduate students and involves key concepts like operator commutativity and expectation.

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