The Open Mapping And Closed Graph Theorems Review

# The Open Mapping and Closed Graph Theorems Review

We will now review some of the recent material regarding the open mapping theorem and closed graph theorem.

# Page Conditions Conclusion
1 Criterion for the Range of a BLO to be Closed when X is a Banach Space 1. $X$ is a Banach space.
2. $Y$ is a normed linear space.
3. $T : X \to Y$ is a bounded linear operator.
4. There exists an $M > 0$ such that for every $y \in T(X)$ there is an $x \in X$ such that $T(x) = y$ and $\| x \| \leq M \| y \|$.
$T(X)$ is a closed subspace of $Y$.
2 Closed Ranges of BLOs when Y is a Banach Space 1. $X$ is a normed linear space.
2. $Y$ is a Banach space.
3. $T : X \to Y$ is a bounded linear operator.
4. $T(X)$ is closed.
There exists an $M > 0$ such that for every $\epsilon > 0$ and every $x \in X$ there exists an $x' \in X$ such that $\| T(x) - T(x') \| < \epsilon$ and $\| x' \| \leq M \| T(x) \|$.
3 IFF Criterion for the Range of a BLO to be Closed when X and Y are Banach Spaces 1. $X$ is a Banach space.
2. $Y$ is a Banach space.
3. $T : X \to Y$ is a bounded linear operator.
$T(X)$ is a closed subspace of $Y$ if and only if there exists an $M > 0$ such that for every $y \in T(X)$ there exists an $x \in X$ such that $T(x) = y$ and $\| x \| \leq M \| y \|$.
4 Second IFF Criterion for the Range of a BLO to be Closed when X and Y are Banach Spaces 1. $X$ is a Banach space.
2. $Y$ is a Banach space.
3. $T : X \to Y$ is a bounded linear operator.
$T(X)$ is a closed subspace of $Y$ if and only if there exists an $M' > 0$ such that for every $x \in X$ with $\| T(x) \| < 1$ there exists an $x' \in X$ such that $T(x) = T(x')$ and $\| x' \| \leq M'$.
5 The Open Mapping Theorem 1. $X$ is a Banach space.
2. $Y$ is a Banach space.
$T : X \to Y$ is a bounded linear operator.
$T(X)$ is a closed subspace of $Y$ if and only if $T$ is an open map.
6 Isomorphisms Between Banach Spaces 1. $X$ is a Banach space.
2. $Y$ is a Banach space.
3. $T : X \to Y$ is a bounded linear operator.
$T$ is an isomorphism from $X$ to $Y$ if and only if $T$ is bijective.

Observe that (3)-(6) have the same conditions/hypotheses.