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.

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