4.1 Open Mapping Theorem: statement, proof, and applications

The Open Mapping Theorem is a cornerstone of functional analysis. It states that surjective bounded linear operators between Banach spaces map open sets to open sets, a powerful result with far-reaching implications.

This theorem's proof relies on the Baire Category Theorem and leads to important applications. It's crucial for establishing other key results like the Closed Graph Theorem and the Inverse Mapping Theorem, shaping our understanding of linear operators.

The Open Mapping Theorem

Open Mapping Theorem statement

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• Asserts that a surjective bounded linear operator $T$ between Banach spaces $X$ and $Y$ is an open map
  • Maps open sets in $X$ to open sets in $Y$
• Requires key assumptions:
  • $X$ and $Y$ are complete normed vector spaces (Banach spaces)
  • $T$ is a continuous linear transformation (bounded linear operator)
  • $T$ is onto, meaning for every $y \in Y$, there exists an $x \in X$ such that $T(x) = y$ (surjective)

Proof using Baire Category Theorem

• Proves by contradiction, assuming $T$ is not an open map
• Supposes an open set $U$ in $X$ exists such that $T(U)$ is not open in $Y$
  • Implies a point $y_0 \in T(U)$ exists where no open ball around $y_0$ is contained in $T(U)$
• Constructs a sequence of closed sets $F_n = \overline{T(nU)}$, where $nU = \{nx : x \in U\}$
  • Each $F_n$ is closed due to $T$'s continuity and $nU$'s closure
• Covers $Y$ with the union of $F_n$, $Y = \bigcup_{n=1}^{\infty} F_n$
  • $T$'s surjectivity ensures for every $y \in Y$, an $x \in X$ exists with $T(x) = y$
  • Scaling $x$ yields an $n$ where $x \in nU$, implying $y \in T(nU) \subseteq F_n$
• Applies the Baire Category Theorem to conclude one $F_n$ must have a non-empty interior
  • Assumes $F_N$ has a non-empty interior, with an open ball $B(y_1, r) \subseteq F_N$ contained in $F_N$
• Deduces $B(y_1, r) \subseteq T(NU)$, contradicting the assumption that no open ball around $y_0$ is contained in $T(U)$
• Concludes $T$ must be an open map

Surjectivity of linear operators

• Establishes that an injective bounded linear operator $T: X \to Y$ between Banach spaces is surjective if and only if $T(X)$ is closed in $Y$
  • Surjectivity implies $T(X) = Y$, which is closed
  • $T(X)$'s closure, combined with the Open Mapping Theorem, implies $T$ is an open map
  • $T$'s injectivity makes it a bijection between $X$ and $T(X)$
  • $T(X)$'s closure makes it a Banach space, and the Inverse Mapping Theorem ensures $T^{-1}$ is continuous
  • $T$ becomes a homeomorphism between $X$ and $T(X) = Y$, implying surjectivity

Applications in functional analysis

• Proves the Closed Graph Theorem
  • A linear operator $T: X \to Y$ between Banach spaces with a closed graph in $X \times Y$ is continuous
• Proves the Inverse Mapping Theorem
  • The inverse of a bijective bounded linear operator $T: X \to Y$ between Banach spaces is also a bounded linear operator
• Proves the Closed Range Theorem
  • For a bounded linear operator $T: X \to Y$ between Banach spaces, $\text{range}(T)$ is closed in $Y$ if and only if $\text{range}(T^*)$ is closed in $X^*$
• Proves the completeness of function spaces like $L^p$ spaces and Sobolev spaces