What Does Norm Convergence Mean in $L^p$ Spaces?

In summary, the theorem states that if $f_n, f \in L^p, 1\leq p < +\infty$, and $f_n \rightarrow f$ almost everywhere, then $f_n\rightarrow f$ as for the norm.
  • #1
mathmari
Gold Member
MHB
5,049
7
Hey! :eek:

If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $||f_n||_p \rightarrow ||f||_p$, then $f_n\rightarrow f$ as for the norm.

Could you give me some hints how to show it?? (Wondering)

What does convergence as for the norm mean?? (Wondering)
 
Physics news on Phys.org
  • #2
mathmari said:
Hey! :eek:

If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $||f_n||_p \rightarrow ||f||_p$, then $f_n\rightarrow f$ as for the norm.

Could you give me some hints how to show it?? (Wondering)

What does convergence as for the norm mean?? (Wondering)
This is an elusive result. I found a proof of it http://math.ucsd.edu/~lni/math240/Hand-out1.pdf (Theorem 0.1). The proof relies on a generalised version of the dominated convergence theorem, which is proved here. Another ingredient in the proof is the fact that $(a+b)^p \leqslant 2^{p-1}(a^p + b^p)$ for positive numbers $a,b$. You can deduce that from the fact that the function $f(x) = x^p$ is convex for $x>0$, and so $f\bigl(\frac12\!(a+b)\bigr) \leqslant \frac12\!\!\bigl(f(a) + f(b)\bigr).$

"Convergence as for the norm" means $\|\,f_n - f\|_p \to0$ as $n\to\infty$.
 
Last edited:
  • #3
Opalg said:
This is an elusive result. I found a proof of it http://math.ucsd.edu/~lni/math240/Hand-out1.pdf (Theorem 0.1). The proof relies on a generalised version of the dominated convergence theorem, which is proved here. Another ingredient in the proof is the fact that $(a+b)^p \leqslant 2^{p-1}(a^p + b^p)$ for positive numbers $a,b$. You can deduce that from the fact that the function $f(x) = x^p$ is convex for $x>0$, and so $f\bigl(\frac12\!(a+b)\bigr) \leqslant \frac12\!\!\bigl(f(a) + f(b)\bigr).$

"Convergence as for the norm" means $\|\,f_n - f\|_p \to0$ as $n\to\infty$.

Ahaa... So can I formulate it as followed?? (Wondering)From Fatou`s lemma we have that $$\int \lim \inf [2^{p-1}(|f_n|^p+|f|_p)-|f_n-f|^p]d\mu \leq \\ \lim \inf \int [2^{p-1}(|f_n|^p+|f|^p)-|f_n-f|^p]d\mu \\ \Rightarrow 2^{p-1}\int \lim \inf (|f_n|^p+|f|^p)d\mu+\int \lim \inf (-|f_n-f|^p)d\mu \leq 2^{p-1}(\lim \inf \int |f_n|^pd\mu +\lim \inf \int |f|^pd\mu )+\lim \inf (-\int |f_n-f|^p d\mu) \\ \Rightarrow 2^{p-1}[\int \lim \inf |f_n |^pd \mu+\int \lim \inf |f|^pd \mu]-\int \lim \sup |f_n-f|^pd \mu \leq 2^{p-1}[\lim \inf \int |f_n|^pd \mu+\lim \inf \int |f|^p d\mu]-\lim \sup \int |f_n-f|^pd \mu \ \ \ \ \ (*) $$

Knowing that $||f_n||_p\rightarrow ||f||_p \Rightarrow \left ( \int |f_n|^p\right )^{1/p}\rightarrow \left ( \int |f|^p\right )^{1p}$ , we have that $\lim \inf |f_n|^p=|f|^p$

Therefore, $$(*)\Rightarrow 2^{p-1}\int (|f|^pd \mu+\int |f|^pd\mu)-\int \lim \sup |f_n-f|^pd \mu \leq 2^{p-1}(\lim \inf \int |f_n|^pd \mu +\int |f|^pd \mu)-\lim \sup \int |f_n-f|^pd \mu \\ \Rightarrow 2^{p-1}\int |f|^pd \mu-\int \lim \sup |f_n-f|^pd \mu \leq 2^{p-1}\lim \inf ||f_n||^p_p-\lim \sup \int |f_n-f|^pd \mu \\ \Rightarrow 2^{p-1}||f||^p_p-\int \lim \sup |f_n-f|^pd \mu \leq 2^{p-1} ||f||^p_p-\lim \sup \int |f_n-f|^pd \mu \\ \Rightarrow \lim \sup \int |f_n-f|^pd \mu \leq \int \lim\sup |f_n-f|^pd \mu \\ \Rightarrow \lim \sup ||f_n-f||^p_p \leq \int \lim \sup |f_n-f|^pd \mu =0, \text{ since } f_n\rightarrow f \text{ almost everywhere } $$

So, we conclude that $||f_n-f||_p\rightarrow 0$.
 
  • #4
Yes, that looks good. (Yes) (Rock)

I struggled for a long time, unsuccessfully, to prove this result by using the dominated convergence theorem. Then I discovered that online reference to a generalisation of the DCT (proved in the same way as the standard DCT, from Fatou's lemma), in which instead of a single dominating function there is a convergent sequence of them.
 
  • #5


Convergence as for the norm means that a sequence of functions, denoted as $f_n$, converges to a function $f$ in the norm defined by $||\cdot||_p$, where $p$ is a real number between 1 and infinity. This norm is a measure of the size or magnitude of a function, and it is defined as the $p$-th root of the integral of the absolute value of the function raised to the power of $p$. In simpler terms, it measures how much the function deviates from zero.

To show that $f_n$ converges to $f$ as for the norm, we need to prove that the norm of the difference between $f_n$ and $f$ approaches zero as $n$ approaches infinity. This can be done by using the triangle inequality and the fact that the norm is a continuous function.

First, we can write $||f_n-f||_p$ as $||f_n-f_n+f-f||_p$. Using the triangle inequality, we can split this into $||f_n-f_n||_p + ||f-f_n||_p$. Since $f_n$ converges to $f$ almost everywhere, we know that $f_n$ and $f$ have the same integral, which implies that $||f_n-f_n||_p = 0$. Therefore, we are left with $||f-f_n||_p$.

Next, using the continuity of the norm, we can write $||f-f_n||_p$ as $||f-f_n||_p \leq ||f-f_n||_p + ||f_n||_p - ||f||_p$. Since $||f_n||_p \rightarrow ||f||_p$, we can choose $n$ large enough such that $||f_n||_p - ||f||_p < \epsilon$, where $\epsilon$ is a small positive number. This means that $||f-f_n||_p \leq ||f-f_n||_p + ||f_n||_p - ||f||_p < \epsilon$. As $n$ approaches infinity, the right-hand side of the inequality approaches zero, which means that $||f-f_n||_p$ also approaches zero. This proves that $f_n$ converges to $f$ as for the norm.

In conclusion, convergence as for
 

FAQ: What Does Norm Convergence Mean in $L^p$ Spaces?

What is convergence as for the norm?

Convergence as for the norm refers to the tendency of data or results from different studies or experiments to align or agree with each other, leading to a consistent or standard conclusion. It is a fundamental principle in science that supports the validity and reliability of research findings.

Why is convergence important in scientific research?

Convergence is essential in scientific research because it allows for the replication and verification of results. When multiple studies converge on the same conclusion, it strengthens the evidence and increases confidence in the findings. It also helps to identify potential errors or biases in individual studies.

What factors can influence convergence in scientific research?

Several factors can impact convergence in scientific research, such as the quality of the study design, the sample size, the statistical methods used, and the validity of the measurements or data collected. The expertise and biases of the researchers involved can also play a role.

How can researchers promote convergence in their studies?

To promote convergence in their studies, researchers can use rigorous and standardized methodologies, carefully select their sample size and participants, and openly share their data and methods with other researchers for replication. They can also conduct meta-analyses, which combine data from multiple studies to assess the overall effect size and level of convergence.

What are the limitations of convergence in scientific research?

While convergence is a crucial aspect of scientific research, it is not always achievable or reliable. Factors such as publication bias, selective reporting of results, and the influence of funding sources can hinder convergence. Additionally, there may be genuine differences in study outcomes due to variations in sample characteristics or study conditions.

Similar threads

Replies
5
Views
495
Replies
9
Views
1K
Replies
4
Views
2K
Replies
11
Views
1K
Replies
21
Views
2K
Replies
2
Views
1K
Replies
6
Views
2K
Replies
13
Views
2K
Back
Top