## Real Analysis Exchange

### Another Proof That Lp-Bounded Pointwise Convergence Implies Weak Convergence

Marian Jakszto

#### Abstract

This note gives another proof of the known fact that $L^{p}$-bounded pointwise convergence implies weak convergence in $L^{p},$ $p>1.$ The proof is based on Banach and Saks’ theorem. The same method applies to convergence in measure.

#### Article information

Source
Real Anal. Exchange, Volume 36, Number 2 (2010), 479-482.

Dates
First available in Project Euclid: 11 November 2011

https://projecteuclid.org/euclid.rae/1321020515

Mathematical Reviews number (MathSciNet)
MR3016731

Zentralblatt MATH identifier
1245.46023

#### Citation

Jakszto, Marian. Another Proof That L p -Bounded Pointwise Convergence Implies Weak Convergence. Real Anal. Exchange 36 (2010), no. 2, 479--482. https://projecteuclid.org/euclid.rae/1321020515

#### References

• S. Banach and S. Saks, Sur la convergence forte dans le espaces ${L}^{p}$, Studia Math., 2 (1930), 51–57.
• E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, 1975.
• E. H. Lieb and M. Loss, Analysis, American Mathematical Society, 2001.
• F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publications, 1990.