$\newcommand{\im}{\mathop{\mathrm{im}}}$

# MA40057 Homework: Hints

Sheet 1  Sheet 2  Sheet 3  Sheet 4  Sheet 5  Sheet 6  Sheet 7  Sheet 8  Sheet 9  Sheet 10

## Sheet 1

### Question 1.(d)

Consider the functions $f_n(x) = \min\{|x|,n\}$.

### Question 2

First establish the case $m = 2$ and then use induction.

### Question 3

In some parts of this questions, the results from the previous parts are useful.

### Question 3.(b)

Use the mean value theorem.

### Question 3.(d)

First fix a partition and prove an inequality for it. Then take the supremum.

### Question 3.(e)

Given a Cauchy sequence, first try to find a candidate for its limit. To this end, use what you know about $C^0([a,b])$.

## Sheet 2

### Question 2.(b)

Two hints here.
1. Use a basis $(e_1,\ldots,e_N)$ of $X$ and work in the corresponding coordinates. That is, for $\alpha = (\alpha_1,\ldots,\alpha_N) \in \mathbb{F}^N$, consider the point $F(\alpha) = \sum_{n = 1}^N \alpha_n e_n$ of $X$. This defines a bijection, and therefore you can work in ${\mathbb F}^N$. However, you need to show that $F$ is continuous.
2. Recall that on a compact set, such as the unit sphere in $\mathbb{F}^N$, a continuous function attains its minimum and its maximum.

### Question 2.(c)

Use the result from (b).

### Question 3

Use the result from 2.(b).

## Sheet 3

### Question 2

Show first that the inner product is continuous.

### Question 3

Use Theorems 2.2.4 and 2.2.5.

### Question 4

Show that $c_0 = \hat{c}_0 := \left\{x = (x_n)_{n \in \mathbb{N}} \in \ell^\infty: \lim_{n \to \infty} x_n = 0\right\}$. To this end, it suffices to show that $\hat{c}_0 \subset c_0$ and $\hat{c}_0$ is closed (justify this claim).

## Sheet 4

### Question 2.(a) and (c)

Consider the sequences $(1,0,0,\ldots)$, $(1,1,0,0,\ldots)$, $(1,1,1,0,0,\ldots)$, etc.

### Question 4.(a)

There is a formula for the radius of convergence.

### Question 4.(b)

Use the theory of power series to study differentiability.

### Question 4.(c)

For $k = 0$ you get a bounded linear operator, for $k = 1$ you don't. To see the latter, choose, e.g., the sequence $(a_n)_{n \in \mathbb{N}}$ with $a_n = \begin{cases} 0 & \text{if n = 1,} \\ (n - 1)^{-3/2} & \text{if n \ge 2.} \end{cases}$ Note: if $f(x) = \sum_{n = 0}^\infty a_n x^n$ for $x \in (-1,1)$, then $f \in C^0([-1,1])$, is not equivalent to convergence of the series at $x = \pm 1$.

## Sheet 5

### Question 2

The graph of $\Pi_U$ consists of all $(x,u) \in X \times U$ such that $x - u \in V$.

### Question 3.(a)

The two norms give rise to two different Banach spaces. Use the open mapping theorem for a suitable operator between them.

### Question 3.(b)

Use (a) and give a proof by contradiction. It may be useful to note that for any $(t_0,\ldots,t_L) \in \Delta$ and $f \in C^1([0,1])$, $\sum_{\ell = 1}^L |f(t_\ell) - f(t_{\ell - 1})| \le \int_0^1 |f'(t)| \, dt.$ If $f$ is increasing, then $\sum_{\ell = 1}^L |f(t_\ell) - f(t_{\ell - 1})| = f(1) - f(0),$ and this helps to compute the norm for suitably constructed functions.

### Question 4

Show first that $P(X) = \{y \in X : Py = y\}$, and if $y = Px$, then $y - x \in \ker P$. Use the closed graph theorem.

## Sheet 6

### Question 2

Consider the functional $\pi_m : \ell^p \to \mathbb{F}$ defined by $\pi_m x = x_m$ for $x = (x_n)_{n \in \mathbb{N}} \in \ell^p$. Use Theorem 4.2.1.

### Question 3.(a)

For $y \in H$ define the functional $g_y \in H^*$ with $g_y(x) = B(x,y)$. Use the Riesz representation theorem.

### Question 3.(b)

Consider $B(y,y)$.

Use part (b).

### Question 3.(d)

Use parts (b) and (c) and Theorem 2.2.1.

### Question 3.(e)

Use the Riesz representation theorem once more.

## Sheet 7

### Question 2.(b)

Consider the function $\theta \mapsto \phi(e^{i\theta} x)$ and use tools from year 2 analysis.

### Question 2.(c)

For $x \in X$ and $y \in S_x$ as in (b), check that $\|y\|_X = \|x\|_X$ and $|\phi(x)| \le |\psi(x)| = |\phi(y)|$.

### Question 3.(a)

Study the proof of Theorem 4.3.1.

### Question 3.(c)

Consider $y = x - \sum_{n = 1}^\infty (x,u_n)u_n$. Recall that $U$ is a complete orthonormal system. What does this mean for $y$?

### Question 4

Consider a Hamel basis $B$ of $X$ and define $f$ by prescribing its values on $B$.

## Sheet 8

### Question 1

First choose a Hamel basis $S$ of $V$ and define the values of $f$ on $S$.

### Question 2.(a)

Consider the linear subspace spanned by $x_0$.

### Question 3

Consider the linear subspace $L = \left\{x = (x_n)_{n \in \mathbb{N}} : \lim_{n \to \infty} x_n \mbox{ exists}\right\}.$ On $L$, consider the functional $f(x) = \lim_{n \to \infty} x_n$.

### Question 4.(b)

If $T$ is injective, then there exists a map $T^{-1} : \im T \to X$ such that $T^{-1} \circ T = \mathrm{id}_X$. For $\phi \in X^*$, the composition $\phi \circ T^{-1}$ is then a functional on $\im T$.
If $T$ is not injective, then for every $x_0 \in \ker T$ and every $\psi \in Y^*$, we have $T^* \psi(x_0) = 0$.

### Question 3.(c)

If $T$ is surjective, then for every $\psi \in Y^*$ and $y \in Y$, there exists an $x \in X$ with $\psi(y) = \psi(Tx)$.
If $T$ is not surjective, prove that for every $y_0 \in Y \backslash \im T$, there exists a $\phi \in Y^*$ with $\psi = 0$ on $\im T$ but $\psi(y_0) \not= 0$.

## Sheet 9

### Question 2.(a)

Don't be discouraged by the abstract nature of these objects. Both sides of the equations are maps $\ell^p \to (\ell^q)^*$. To evaluate them, take an $x \in \ell^p$ and plug it in. This gives you an element of $(\ell^q)^*$, so take a $y \in \ell^q$ and plug it in. Now use the definitions and calculate.

### Question 2.(b) – (d)

Use the results from questions 8.3 and 8.4 and Theorem 4.2.1.

### Question 3

Rewrite (i) in terms of the inner product. For any $x_0 \in H$, the functional $x \mapsto (x,x_0)$ belongs to $H^*$.

### Question 4

Use the Riesz representation theorem and Bessel's inequality.

## Sheet 10

### Question 1.(b)

Use Corollary 4.4.1.

### Question 2

Consider a sequence $(x_n)_{n \in \mathbf{N}}$ with $x_n \in C_n$ for every $n \in \mathbf{N}$.

### Question 3 and Question 4.(a)

Eigenvalues can be characterised in terms of the solvability of a certain equation. When you work in $\ell^p$, then that equation gives an infinite system of linear equations. Under what conditions can you solve it? Once you have found a solution, don't forget to check that it gives rise to a point in $\ell^p$.

### Question 4.(b)

Use Theorems 6.1.1 and 6.1.2 and use the information from part (a).

### Question 5

Use questions 2.3, 5.2, and 5.4. Use also Riesz's lemma (Theorem 1.3.3) as in the proof of Theorem 1.3.4.