**Exercise 4.1** Let \(\mathbb{R}\) be given the standard topology. Show that the closed set [0,1] is a \(G_\delta\)-set. Show that the open set \((0,1)\) is an \(F_\sigma\)-set.

**Exercise 4.2** Show that the space \(C[0,1]\) of continuous functions on the closed interval [0,1] is not complete in the norm

$$\|f\|_2=\left(\int_0^1 |f(s)|^2\, ds\right)^{1/2},\quad f\in C[0,1].$$

(The completion of \(C[0,1]\) in the norm \(\|\cdot\|_2\) is \(L_2(0,1)\), by Lusin’s Theorem.)

**Exercise 4.3** Let *M* and *E* be complete metric spaces. Suppose \(h:M\rightarrow E\) is a homeomorphism onto its image (i.e., *h* is a continuous one-to-one map, and \(h^{-1}|_{h(M)}\) is continuous). Show that *h*(*M*) is a \(G_\delta\)-set.

**Exercise 4.4** Let *X* be a Banach space and suppose *E* is a dense linear subspace which is a \(G_\delta\)-set. Show that *E = X*.

**Exercise 4.5** Show that if *Y* is a normed space which is homeomorphic to a complete metric space, then *Y* is a Banach space. (*Hint:* Consider *Y* as a dense subspace in its completion.)

**Exercise 4.6** Let *X* and *Y* be Banach spaces and let \(T:X\rightarrow Y\) be a bounded linear operator. If *M* is a closed subspace of *X*, show that either *T*(*M*) is first category in *Y* or \(T(M)=Y\).

**Exercise 4.7** Let \(X=C^{(1)}[0,1]\) be the space of continuously differentiable functions on [0,1] and let \(Y=C[0,1]\). Equip both spaces with the supremum norm \(\|\cdot\|_\infty\). Define a linear map \(T:X\rightarrow Y\) by \(T(f) = f'\) for all functions \(f\in C^{(1)}[0,1]\). Show that *T* has closed graph, but *T* is not continuous. Conclude that \(\left(C^{(1)}[0,1], \|\cdot\|_\infty\right)\) is not a Banach space.

**Exercise 4.8** Let \(\phi \in C[0,1]\) be a function which is not identically 0. Show the set \(M=\{\phi \, f: f\in C[0,1]\}\) is of the first category in \(C[0,1]\) if and only if \(\phi(x)=0\) for some \(x\in [0,1]\).

**Exercise 4.9** Let \(\displaystyle (a_k)_{k\in\mathbb{Z}}\) be a sequence of complex scalars with only finitely many nonzero terms. Define a trigonometric polynomial \(f:\mathbb{T}\rightarrow\mathbb{C}\) by \(\displaystyle f(\theta) = \sum_{k\in\mathbb{Z}} a_k\, e^{ik\theta}.\) Show that \(\hat{f}(n) = a_n\) for all \(n\in\mathbb{Z}\).

**Exercise 4.10** Show that there exists a function \(f\in L_1(\mathbb{T})\) whose Fourier series fails to converge to *f* in the *L* _{1}-norm. Precisely, show that if

$$S_N f = \sum_{k=-N}^N \hat{f}(k)\, e^{ik\theta},$$

then there exists an \(f\in L_1(\mathbb{T})\) such that \(\|f - S_Nf\|_{L_1(\mathbb{T})}\) does not tend to 0.

**Exercise 4.11** Show that the Cesàro means \(\frac{1}{N}(S_1 f + \cdots + S_N f)\) converge to *f* in the *L* _{1}-norm for every \(f\in L_1(\mathbb{T})\).

**Exercise 4.12** Let *X* and *Y* be Banach spaces. If \(T:X\rightarrow Y\) is a bijection, show that the adjoint map \(T^\ast:Y^\ast\rightarrow X^\ast\) is also a bijection. Conclude that if *T* is an isomorphism of Banach spaces, then so is \(T^\ast\).

**Exercise 4.13** Show that the Hilbert transform of Example 4.40 is well-defined.

**Exercise 4.14** Let \(f:[1,\infty)\rightarrow\mathbb{R}\) be a continuous function. Suppose \((\xi_n)_{n=1}^\infty\) is a strictly increasing sequence of real numbers with \(\xi_1\geq 1\), \(\displaystyle\lim_{n\rightarrow\infty} \xi_n = \infty\), and \(\displaystyle\lim_{n\rightarrow\infty} \frac{\xi_{n+1}}{\xi_n}=1\). If \(\displaystyle\lim_{n\rightarrow\infty} f(\xi_n x) = 0\) for all \(x\geq 1\), then prove that \(\displaystyle\lim_{x\rightarrow\infty} f(x) = 0\).

**Exercise 4.15** Show that \(L_2(0,1)\) is of the first category in \(L_1(0,1)\).

**Exercise 4.16** Let \((\Omega,\mu)\) be a probability space and suppose there exists a sequence of disjoint sets \((E_n)_{n=1}^\infty\) such that \(\mu(E_n)>0\) for all \(n\in\mathbb{N}\). Show that \(L_p(\Omega,\mu) \neq L_q(\Omega,\mu)\) if \(1 \leq p < q < \infty\).

**Exercise 4.17** Let *X* be an infinite-dimensional Banach space and suppose *V* is a closed subspace of *X*. If both *V* and \(X/V\) are separable, then show *X* is separable.

**Exercise 4.18** Identify the quotient space \(c/c_0\).

**Exercise 4.19** Let \(1\leq p \leq \infty\) and let \(V=\{(x_j)_{j=1}^\infty\in\ell_p: x_{2k}=0 \mbox{ for all} k\in\mathbb{N}\}\). Show that \(\ell_p/V\) is isometrically isomorphic to *ℓ* _{ p }.

**Exercise 4.20** Find an example of a map \(T:X\rightarrow Y\), where *X* and *Y* are normed spaces, such that *T* is a bounded linear bijection, but *T* ^{-1} is not bounded. (*Hint: X* and *Y* cannot both be Banach spaces, or *T* ^{-1} will be bounded by Corollary 4.30.)

**Exercise 4.21** Show that the Closed Graph Theorem (Theorem 4.35) implies the Open Mapping Theorem (Theorem 4.29).