Does L^2 Convergance Imply Convergance of L^2 norms?

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E((X_n - X)^2)} \sqrt{E(X^2)}\to 0In summary, The result can be proven using the Cauchy-Bunyakovski-Schwarz inequality, which states that if E((X_n-X)^2)\to0, then E(X_n^2)\to E(X^2). This result can be found in a book or using a reference.
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The answer seems to obviously be yes. But it's not so obvious to show it.

I'm working with random variables. So the [itex]L^2[/itex] norm of X is [itex]E(X^2)^{1/2}[/itex], where E is the expected value. Thus, we want to show: if [itex]E((X_n-X)^2)\to0[/itex], then [itex]E(X_n^2)\to E(X^2)[/itex].

From [itex]E((X_n^2-X)^2)\to0[/itex], we get
[tex]E(X_n^2)\to2E(X_nX)-E(X^2).[/tex]

I think it should be true that [itex]2E(X_nX)\to2E(X^2)[/itex], which would prove the result, but I'm not sure how to prove that.

Any help?

Or a reference? Is the result in a book?
 
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Use the Cauchy-Bunyakovski-Schwarz inequality

[tex]|E(X_n X) - E(X^2)| = |E( (X_n - X) X)|\leq \sqrt{ E((X_n - X)^2) E(X^2)}[/tex]
 
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1. What is L^2 convergence?

L^2 convergence refers to a type of convergence in mathematics where a sequence of functions or vectors in a Hilbert space approach a limit in the L^2 norm. This means that the square root of the sum of squared magnitudes of the sequence approaches zero as the sequence progresses.

2. What is the difference between L^2 convergence and pointwise convergence?

The main difference between L^2 convergence and pointwise convergence is that L^2 convergence considers the convergence of a sequence as a whole, while pointwise convergence considers the convergence of individual elements in a sequence. L^2 convergence is a stronger form of convergence compared to pointwise convergence.

3. Does L^2 convergence imply convergence of L^2 norms?

Yes, L^2 convergence implies convergence of L^2 norms. This is because the L^2 norm is defined as the square root of the sum of squared magnitudes, which is the same as the definition of L^2 convergence. Therefore, as the sequence converges in the L^2 norm, it also converges in the L^2 convergence sense.

4. What are some applications of L^2 convergence?

L^2 convergence has various applications in mathematics, physics, and engineering. It is used in functional analysis, signal processing, and image processing, among others. It is also used in the study of Fourier series and integral transforms.

5. Are there any conditions for L^2 convergence to imply convergence of L^2 norms?

Yes, there are some conditions for L^2 convergence to imply convergence of L^2 norms. One such condition is that the underlying space should be a Hilbert space. Additionally, the sequence in question should satisfy certain properties, such as being Cauchy or uniformly bounded. These conditions ensure the validity of the implication.

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