# Closed subspace of a Sobolev Space

• lmedin02
In summary, the conversation discusses the space \tilde{W}^{1,2}(\Omega) and its properties, specifically that it is a closed subspace of W^{1,2}(\Omega). The conversation also mentions that W^{1,2}(\Omega) is the space of L^2(\Omega) with distributional derivatives also in L^2(\Omega). The main focus of the conversation is on showing that a convergent sequence in \tilde{W}^{1,2}(\Omega) has an average value of 0 and therefore belongs in \tilde{W}^{1,2}(\Omega). The question of whether the sequence needs to be bounded is also brought up.
lmedin02

## Homework Statement

I am considering the space $\tilde{W}^{1,2}(\Omega)$ to be the class of functions in $W^{1,2}(\Omega)$ satisfying the property that its average value on $\Omega$ is 0. I would like to show that $\tilde{W}^{1,2}(\Omega)$ is a closed subspace of $W^{1,2}(\Omega)$.

## Homework Equations

$W^{1,2}(\Omega)$ is the space of $L^2(\Omega)$ so that their distributional derivative also lie in $L^2(\Omega)$.

## The Attempt at a Solution

It is clear that $\tilde{W}^{1,2}(\Omega)$ is a subspace of ${W}^{1,2}(\Omega)$. So I now consider a convergent sequence of functions $\tilde{w}_k$ in $\tilde{W}^{1,2}(\Omega)$ converging to a function $w$ in $W^{1,2}(\Omega)$. I am having trouble showing that $w$ has average value 0 and hence belongs in $\tilde{W}^{1,2}(\Omega)$. Any suggestions.

I am having a hard time making this conclusion. How about if the sequence is bounded?

## What is a closed subspace of a Sobolev Space?

A closed subspace of a Sobolev Space is a subset of a Sobolev Space that is itself a complete vector space under the same norm as the Sobolev Space. This means that it contains all of its limit points, making it "closed" in a topological sense.

## What are the properties of a closed subspace of a Sobolev Space?

A closed subspace of a Sobolev Space inherits all of the properties of the larger Sobolev Space, such as linearity and continuity. It also has the additional property of being closed, which means that it is complete and contains all of its limit points.

## How is a closed subspace of a Sobolev Space related to the Sobolev Space?

A closed subspace is a subset of a Sobolev Space, meaning that it is contained within the Sobolev Space. It shares the same norm as the Sobolev Space, but has the additional property of being closed, making it a complete vector space.

## What is the significance of a closed subspace in Sobolev Spaces?

A closed subspace is useful in Sobolev Spaces because it allows for the construction of unique solutions to partial differential equations. By working within a closed subspace, we can ensure that our solutions are complete and contain all of their limit points, avoiding any issues with non-uniqueness.

## How can a closed subspace be constructed in Sobolev Spaces?

A closed subspace can be constructed by taking a subset of functions from the larger Sobolev Space that satisfy certain boundary or initial conditions. These conditions can be imposed using appropriate function spaces, such as the spaces of continuous or smooth functions, to ensure that the subspace is complete and closed.

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