Banach-Steinhaus Theorem

(Part of the proof of Theorem 7.1). Let $X$ be a Banach space, $Y$ a normed space, and $\left\{ T_{n}\right\}_{n = 1}^{\infty } \subset \mathcal{B}(X, Y)$, with $\left\| T_{n}\right\| \rightarrow \infty$ as $n \rightarrow \infty$. Prove that for any $\lambda > 0$ the set

is nowhere dense in $X$.

Theorem 7.1 (S. Banach, H. Steinhaus, 1927): let $X$ be a Banach space, $Y$ a normed space, and $\mathcal{M} \subset \mathcal{B}(X, Y)$ a family of operators. If for every $x \in X$ $\sup_{A \in \mathcal{M}}\left\| A x\right\| \leqslant C(x)$, where the constant $C(x)$ depends only on the vector $x$, then there exists a constant $C_{1} > 0$ such that $\sup_{A \in \mathcal{M}}\left\| A\right\| \leqslant C_{1}$, i.e. the set $\mathcal{M}$ is bounded in the space $\mathcal{B}(X, Y)$.

Show that the completeness of the space $X$ in the statement of Theorem 7.1 is essential.

Theorem 7.1 (S. Banach, H. Steinhaus, 1927): let $X$ be a Banach space, $Y$ a normed space, and $\mathcal{M} \subset \mathcal{B}(X, Y)$ a family of operators. If for every $x \in X$ $\sup_{A \in \mathcal{M}}\left\| A x\right\| \leqslant C(x)$, where the constant $C(x)$ depends only on the vector $x$, then there exists a constant $C_{1} > 0$ such that $\sup_{A \in \mathcal{M}}\left\| A\right\| \leqslant C_{1}$, i.e. the set $\mathcal{M}$ is bounded in the space $\mathcal{B}(X, Y)$.

Let $f_{n} \in l_{\infty }$ be a sequence of functionals on the space $l_{1}, f_{n}(x) = n x_{n}, n = 1,2, \ldots$ This family of functionals is not bounded in the space $l_{\infty } = \left(l_{1}\right)^{*}$, and hence, by the Banach-Steinhaus theorem, there is a vector $x \in l_{1}$ such that $\sup \left|f_{n}(x)\right| = \infty$. Find such a vector $x$.

(E. Hellinger, O. Toeplitz, 1910). Let $A: H \rightarrow H$ be a linear operator in a Hilbert space $H$. Prove that if for all $x, y \in H$ the equality $(A x, y) = (x, A y)$ holds, then the operator $A$ is bounded. ¹

Definition 8.3: An operator $A \in \mathcal{B}(H)$ is called self-adjoint if $A = A^{*}$.

Let $X$ and $Y$ be Banach spaces. Prove that if a bilinear form $B(x, y)$, defined on $X \times Y$, is continuous separately in each argument, then it is continuous jointly in both arguments.

Define the operator

in the space $C[-\pi , \pi ]$. Prove that $\left\| S_{n}\right\| \rightarrow \infty$. Deduce from this that there exists a continuous function $x \in C[-\pi , \pi ]$ whose trigonometric Fourier series diverges at the point $t = 0$.