The Closed Graph Theorem

The Closed Graph Theorem

Definition: Let $f : X \to Y$ be a function. Then the Graph of $f$ is defined to be $\mathrm{Gr} (f) = \{ (x, f(x)) : x \in X \}$.
Theorem 1 (The Closed Graph Theorem): Let $X$ and $Y$ be Banach spaces and let $T : X \to Y$ be a linear operator. Then $T$ is a bounded linear operator if and only of the graph of $T$, $\mathrm{Gr} (T) = \{ (x, T(x)) : x \in X \}$ is a closed set in the product space $X \times Y$, that is, if $(x_n)_{n=1}^{\infty}$ is a sequence of points in $X$ that converges to $x \in X$ and $(T(x_n))_{n=1}^{\infty}$ converges to $y \in Y$ then $T(x) = y$.
  • Proof: $\Rightarrow$ Suppose that $\mathrm{Gr} (T)$ is closed. Then if $(x_n, T(x_n))_{n=1}^{\infty}$ is a sequence of points in the graph of $T$ that converges, it converges to some point $(x, y) \in \mathrm{Gr} (T)$ (since a closed set contains all of its accumulation points). In other words, for every sequence of points $(x_n)_{n=1}^{\infty}$ that converges to $x \in X$ and $(T(x_n))_{n=1}^{\infty}$ that converges to $y \in Y$ we have that $(x, y) \in \mathrm{Gr} (T)$, i.e., $T(x) = y$.
  • $\Leftarrow$ Suppose that $\mathrm{Gr}(T)$ is a closed set in the product space $X \times Y$. Define a new norm $\| \cdot \|_T$ on $X$ for all $x \in X$ by:
(1)
\begin{align} \quad \| x \|_T = \| x \| + \| T(x) \| \end{align}
  • We must first verify that $\| \cdot \|_T$ is indeed a norm.
  • Suppose that $\| x \|_T = 0$. Then $\| x \| + \| T(x) \| = 0$. Since $\| x \| \geq 0$ and $\| T(x) \| \geq 0$ we must have that $\| x \| = 0$. But $\| \cdot \|$ is a norm so $x = 0$. Now we also have that $\| 0 \|_T = \| 0 \| + \| T(0) \| = 0 + 0 = 0$. So $\| x \|_T = 0$ if and only if $x = 0$.
  • Let $\lambda \in \mathbb{C}$. Then:
(2)
\begin{align} \quad \| \lambda x \|_T = \| \lambda x \| + \| T(\lambda x) \| = | \lambda | \| x \| + | \lambda | \| T(x) \| = | \lambda |(\| x \| + \| T(x) \|) \end{align}
  • Lastly, let $x, y \in X$. Then:
(3)
\begin{align} \quad \| x + y \|_T = \| x + y \| + \| T(x + y) \| \leq \| x \| + \| y \| + \| T(x) \| + \| T(y) \| = (\| x \| + \| T(x) \|) + (\| y \| + \| T(y) \|) = \| x \|_T + \| y \|_T \end{align}
  • So indeed, $\| \cdot \|_T$ is a norm on $X$. Furthermore, since $\| x \| \geq 0$ and $\| T(x) \| \geq 0$ we have for all $x \in X$ that:
(4)
\begin{align} \quad \| x \| \leq \| x \| + \| T(x) \| = 1 \cdot \| x \|_T \quad (*) \end{align}
(5)
\begin{align} \quad \| T(x) \| \leq \| x \| + \| T(x) \| = 1 \| x \|_T \quad (**) \end{align}
  • We aim to show that $X$ with the norm $\| \cdot \|_T$ is a Banach space. Let $(x_n)_{n=1}^{\infty}$ be a Cauchy sequence in $X$ with respect to the norm $\| \cdot \|_T$. Let $m, n \in \mathbb{N}$. Then from $(*)$ and $(**)$ we have that:
(6)
\begin{align} \quad \| x_m - x_n \| \leq \| x_m - x_n \|_T \quad (***) \end{align}
(7)
\begin{align} \quad \| T(x_m) - T(x_n) \| \leq \| x_m - x_n \|_T \quad (****) \end{align}
  • From $(***)$ we have that $(x_n)_{n=1}^{\infty}$ must be a Cauchy sequence in $X$ with respect to the original norm on $X$. And from $(****)$ we have that $(T(x_n))_{n=1}^{\infty}$ must be a Cauchy sequence in $Y$ with respect to the norm on $Y$.
  • Since $X$ is a Banach space, $(x_n)_{n=1}^{\infty}$ converges to some $x \in X$ and since $Y$ is a Banach space, $(T(x_n))_{n=1}^{\infty}$ converges to some $y \in Y$.
  • So $(x_n)_{n=1}^{\infty}$ is a sequence of points in $X$ that converges to $x \in X$ and $(T(x_n)_{n=1}^{\infty}$ converges to $y \in Y$, so $T(x) = y$.
  • Now all that remains to show is that the Cauchy sequence $(x_n)_{n=1}^{\infty}$ in $X$ converges to $x \in X$ with respect to the $\| \cdot \|_T$] norm. We have that:
(8)
\begin{align} \quad \lim_{n \to \infty} \| x_n - x \|_T = \lim_{n \to \infty} [\| x_n - x \| + \| T(x_n) - T(x) \|] = \lim_{n \to \infty} \| x_n - x \| + \lim_{n \to \infty} \| T(x_n) - y \| = 0 + 0 = 0 \end{align}
  • So indeed every Cauchy sequence $(x_n)_{n=1}^{\infty}$ in $X$ converges to some $x \in X$ with respect to the $\| \cdot \|_T$ norm. So $X$ is a Banach space with respect to the $\| \cdot \|_T$ norm.
  • From $(*)$ we have that $\| \cdot \|$ and $\| \cdot \|_T$ are equivalent norms by the theorem on the Equivalence of Norms on Banach Spaces page. So there exists a $C > 0$ such that for all $x \in X$:
(9)
\begin{align} \quad \| x \|_T \leq C \| x \| \end{align}
  • But from $(**)$ this means that for all $x \in X$:
(10)
\begin{align} \quad \| T(x) \| \leq C \| x \| \end{align}
  • So $T$ is a bounded linear operator. $\blacksquare$
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