Fundamental Theorems

Show that $L^{2}[0,1]$ is a subset of the first category in $L^{1}[0,1]$.

Let $X \neq \left\{ 0\right\}$ be a normed space, and $A \subset X$ which is not nowhere dense. We denote by $A^{\prime }$ the derivative set of $A$, i.e., the set of all accumulation points of A. Prove that

(Note: $\bar{A} = A \cup A^{\prime }$).

Let $X$ is a Banach space and $A \subset X$ a dense set. Can we find a function $f: X \rightarrow \mathbb {R}$ such that, for every $x \in A$, we have $\lim_{t \rightarrow x}\left|f(t)\right| = \infty$?

Show that $\mathbb {Q}$ is a subset of the first category of $\mathbb {R}$.

Show that the set of piecewise linear functions on $\mathbb {R}$ is of the first category.

Let $X$ and $Y$ be Banach spaces and $T \in \mathcal{B}(X, Y)$. Suppose $T$ is bijective. Show that there exist real numbers $a, b > 0$ such that

Show that the the graph of $T$ is closed but $T$ is not bounded. Does this contradict the closed graph theorem?

Let $H$ be a separable Hilbert space with an orthonormal basis $\left\{ e_{n}\right\}_{n = 1}^{\infty }$. Let $\left\{ x_{n}\right\}$ be a sequence in $H$. Prove that the following two statements are equivalent:

1. $\lim_{n \rightarrow \infty } \left\langle x, x_{n} \right\rangle = 0, \quad \forall x \in H$.

2. $\lim_{n \rightarrow \infty } \left\langle e_{m}, x_{n} \right\rangle = 0, \quad \forall m \in \mathbb {N}$ and $\left\{ \left\| x_{n}\right\| \right\}$ is bounded.

Let $H$ be a Hilbert space and $T: H \rightarrow H$ a linear operator which is symmetric, i.e.,

Prove that $T$ is continuous.

Let $a = \left(a_{1}, a_{2}, \ldots \right) = \left(a_{n}\right)_{n \in \mathbb {N}}$ be a sequence of scalars such that the sequence $\left(a_{n} x_{n}\right)_{n \in \mathbb {N}} \in c_{0}$ for all sequences $x = \left(x_{n}\right)_{n \in \mathbb {N}} \in c_{0}$. Prove that $a \in \ell^{\infty }$.

Let $a = \left(a_{1}, a_{2}, \ldots \right) = \left(a_{n}\right)_{n \in \mathbb {N}}$ be a sequence of scalars such that the sequence $\left(a_{n} x_{n}\right)_{n \in \mathbb {N}} \in c_{0}$ for all sequences $x = \left(x_{n}\right)_{n \in \mathbb {N}} \in \ell^{\infty }$. Prove that $a \in c_{0}$.

Let $x = \left(x_{1}, x_{2}, \ldots \right) = \left(x_{i}\right)_{i = 1}^{\infty }$ be a sequence of scalars such that the series $\sum_{i = 1}^{\infty } x_{i} y_{i}$ is convergent for all $y = \left(y_{1}, y_{2}, \ldots \right) \in c_{0}$. Prove that $x \in \ell^{1}$.

Let $c = \left(c_{1}, c_{2}, \ldots \right) = \left(c_{i}\right)_{i = 1}^{\infty }$ be a sequence of scalars such that the series $\sum_{i = 1}^{\infty } c_{i} a_{i}$ is convergent for all $a = \left(a_{1}, a_{2}, \ldots \right) \in \ell^{1}$. Prove that $x \in \ell^{\infty }$.

Let $X, Y$ and $Z$ be Banach spaces. Suppose that $T: X \rightarrow Y$ is linear, that $J: Y \rightarrow Z$ is linear, bounded and injective, and that $J T \equiv J \circ T: X \rightarrow Z$ is bounded. Show that $T$ is also bounded.

Let $X$ be a Banach space and $E, F$ two closed subspaces of $X$ such that $X = E \oplus F$. Consider the projections on $E$ and on $F$ defined by

Use the closed graph theorem to show that $P_{E} \in \mathcal{B}(X, E)$ and $P_{F} \in \mathcal{B}(X, F)$.

Let $\left(X,\left\| .\right\|_{1}\right)$ and $\left(X,\left\| .\right\|_{2}\right)$ be Banach spaces. Suppose that

Show that the two norms $\left\| .\right\|_{1}$ and $\left\| .\right\|_{2}$ are equivalent.