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There are no doubts that open mapping theorem, closed graph theorem, bounded inverse theorem, uniform boundedness principle are the fundamental theorems of functional analysis. All of them are similar in the sense that they explicitly or implicitly use the Baire category theorem, but in the details they are different. However, all these theorems can be formulated as continuity of certain seminorms defined on a normed spaces.

The proof of the lemma is very similar to the usual proof by Banach-Schauder of the open mapping theorem and is actually stronger than the usual consequences of the Baire category theorem in basic functional analysis. The lemma can be used to easily prove the aforementioned theorems.

I would like to thank the user t.b. for drawing our attention to this lemma and adaptation of the original proof from the general case of linear metric spaces, which he gave as an answer to the question Direct Approach to the Closed Graph Theorem on the main site.

Lemma. (Zabreiko, [1], [2]) Let \(X\) be a Banach space and let \(p: X \to [0,\infty)\) be a seminorm. If for all absolutely convergent series \(\sum_{n=1}^\infty x_n\) in \(X\) we have \[ p\left(\sum_{n=1}^\infty x_n\right) \leq \sum_{n=1}^\infty p(x_n) \in [0,\infty] \] then \(p\) is continuous. That is to say, there exists a constant \(C \geq 0\) such that \(p(x) \leq C\lvert x\rvert _{X}\) for all \(x \in X\).

Proof. Let \(A_n = p^{-1}([0,n])\) and \(F_n = \overline{A_n}\). Note that \(A_n\) and \(F_n\) are symmetric and convex because \(p\) is a seminorm. We have \(X = \bigcup_{n=1}^\infty F_n\) and Baire’s theorem implies that there is \(N\) such that the interior of \(F_N\) is nonempty.

Therefore, there are \(x_0 \in X\) and \(R \gt 0\) such that \(B_R(x_0) \subset F_N\). By symmetry of \(F_N\) we have \(B_{R}(-x_0) = -B_{R}(x_0) \subset F_n\), too. If \(\lvert x\rvert \lt R\) then \(x+x_0 \in B_{R}(x_0)\) and \(x-x_0 \in B_{R}(-x_0)\), so \(x \pm x_0 \in F_{N}\). By convexity of \(F_N\) it follows that \[ x = \frac{1}{2}(x-x_0) + \frac{1}{2}(x+x_0) \in F_N, \] so \(B_R(0) \subset F_N\). But, in fact, we can prove much more: \[ B_{R}(0) \subset A_N \] Suppose \(\lvert x\rvert \lt R\) and choose \(r\) such that \(\lvert x\rvert \lt r \lt R\). Fix \(0 \lt q \lt 1-\frac{r}{R}\), so \(\frac{1}{1-q} \frac{r}{R} \lt 1\). Then \(y = \frac{R}{r}x \in B_{R}(0) \subset F_N = \overline{A_N}\), so there is \(y_{0} \in A_N\) such that \(\lvert y-y_0\rvert \lt qR\), so \(q^{-1}(y-y_0) \in B_R\). Now choose \(y_1 \in A_N\) with \(\lvert q^{-1}(y-y_0) – y_1\rvert \lt q R\), so \(\lvert (y-y_0 – qy_1)\rvert \lt q^2 R\). By induction we obtain a sequence \( (y_k)\subset A_N\) such that \[ \left\lvert y – \sum_{k=0}^n q^k y_k\right\rvert \lt q^n R \quad \text{for all }n \geq 0, \] hence \(y = \sum_{k=0}^\infty q^k y_k\). Observe that by construction \(|y_k| \leq R + qR\) for all \(k\), so the series \(\sum_{k=0}^\infty q^k y_k\) is absolutely convergent. But then the countable subadditivity hypothesis in \(p\) implies that \[ p(y) = p\left(\sum_{k=0}^\infty q^k y_k\right) \leq \sum_{k=0}^\infty q^k p(y_k) \leq \frac{1}{1-q} N \] and thus \(p(x) \leq \frac{r}{R} \frac{1}{1-q} N \lt N\) which means \(x \in A_N\), as we wanted.

Finally, for any \(x \neq 0\), we have with \(\lambda = \frac{R}{\lvert x\rvert (1+\varepsilon)}\) that \(\lambda x \in B_{R}(0) \subset A_N\), so \(p(\lambda x) \leq N\) and thus \(p(x) \leq \frac{N(1+\varepsilon)}{R} \lvert x\rvert \), as desired. \(\blacksquare\)

Now the proof of the fundamental theorems becomes an easy exercise.

1. The open mapping theorem. Hint: set \(p(y) = \inf_{x\in T^{-1}(y)}\lvert x\rvert \).
2. The bounded inverse theorem. Hint: set \(p(x) = \lvert T^{-1}(x)\rvert \).
3. The uniform boundedness principle. Hint: set \(p(x) = \sup_{i\in I}\lvert T_i (x)\rvert \).
4. The closed graph theorem. Hint: set \(p(x) = \lvert T(x)\rvert \).

Detailed solutions can be found in theorems \(1.6.5\), \(1.6.6\), \(1.6.9\) and \(1.6.11\) in An Introduction to Banach Space Theory Graduate Texts in Mathematics. Robert E. Megginson.

[1] П. П. Забрейко, Об одной теореме для полуаддитивных функционалов, Функциональный анализ и его приложения, 3:1 (1969), 86–88

[2] P. P. Zabreiko, A theorem for semiadditive functionals, Functional analysis and its applications 3 (1), 1969, 70-72)