Weakly convergent sequences are bounded

• lmedin02
In summary, the conversation discusses the attempt to show that a weakly convergent sequence is necessarily bounded. The speaker considers a sequence in X'' and uses the uniform boundedness principle or the Banach-Steinhaus Theorem to make their conclusion. They also mention using Hahn-Banach to find a functional that attains the norm of the sequence at a specific point.
lmedin02

Homework Statement

I would like to show that a weakly convergent sequence is necessarily bounded.

The Attempt at a Solution

I would like to conclude that if I consider a sequence ${Jx_k}$ in $X''$. Then for each $x'$ in $X'$ we have that $\sup|Jx_k(x')|$ over all $k$ is finite. I am not sure why this is the case since
$\sup|Jx_k(x')|=\sup|x'(x_k)|\leq\sup||x'||||x_k||$ and $\sup||x_k||$ over all $k$ does not seem finite.

From here I would be able to make my conclusion by applying the uniform boundedness principle or the Banach-Steinhaus Theorem.

I don't really understand your notation.

Shouldn't you just look at various functionals applied to your sequence - since each of these will converge, you get pointwise boundedness. But then apply Hahn-Banach to find a functional that attains the norm of your sequence at a specific point. So...

Thanks for your comments. I was able to resolve my issue with this problem.

What does it mean for a sequence to be weakly convergent?

Weak convergence is a concept in mathematics that describes the behavior of a sequence of numbers as its elements approach a limiting value. In this case, the sequence is said to be weakly convergent if its elements converge to a limit point in a specific mathematical space.

What is the difference between weak convergence and strong convergence?

The main difference between weak and strong convergence lies in the type of limiting values that the sequence approaches. In strong convergence, the sequence approaches a single, unique limit point. In weak convergence, the sequence may approach multiple limit points, with the limit points being different for different subsequences.

Why is it important for weakly convergent sequences to be bounded?

Boundedness is a key property of weakly convergent sequences because it ensures the existence of a limit point. If a sequence is unbounded, it may not have a limit point, and therefore cannot be considered weakly convergent. Boundedness also allows for the application of certain mathematical theorems and techniques to prove convergence.

Can a sequence be weakly convergent but not bounded?

No, a sequence cannot be weakly convergent if it is not bounded. As mentioned previously, boundedness is a necessary condition for weak convergence. If a sequence is unbounded, it cannot have a limit point, and therefore cannot be considered weakly convergent.

How can we prove that a sequence is weakly convergent and bounded?

There are several methods for proving that a sequence is weakly convergent and bounded, such as using the definition of weak convergence or using mathematical theorems such as the Banach-Alaoglu theorem. Additionally, it may be helpful to use techniques such as the Bolzano-Weierstrass theorem to show that the sequence is bounded, and then use the definition of weak convergence to prove convergence to a limit point.

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