Homework 1

a) The set $C^\infty(\mathbb{R})$, which is the set of all functions from $\mathbb{R}$ to $\mathbb{R}$ that are infinitely differentiable, together with the operations of addition and scalar multiplication defined as follows, is a vector space: For any $f_1, f_2 \in C^\infty(\mathbb{R})$

$(f_1 + f_2)(x) = f_1(x) + f_2(x), \qquad x \in \mathbb{R}$

and for any $\alpha \in \mathbb{R}$ and $f \in C^\infty(\mathbb{R})$

$(\alpha \cdot f)(x) = \alpha f(x), \qquad x \in \mathbb{R}$

i) Now, consider $\mathcal{P}(\mathbb{R})$ to be the set of all (polynomial) functions that maps $\mathbb{R}$ to $\mathbb{R}$ such that any $f \in \mathcal{P}(\mathbb{R})$ can be written as $f(x) = \sum_{i=0}^{n} a_i x^i$ for some $n \in \mathbb{N}$ with $a_0, a_1, \cdots, a_n \in \mathbb{R}$. Suppose that we define the same addition and scalar multiplication operations as defined above. Is $\mathcal{P}(\mathbb{R})$ a subspace in $C^\infty(\mathbb{R})$?

ii) Show that the space of all functions in $C^\infty(\mathbb{R})$ which map $\mathbb{R}$ to $\mathbb{R}$ which satisfy $f(10) = 0$ is a vector space with addition and multiplication defined as above.

b) Consider the set $\mathbb{R}^n$. On $\mathbb{R}^n$, define an addition operation and a scalar multiplication operation as follows:

$(x_1, x_2, \cdots, x_n) + (y_1, y_2, \cdots, y_n) = (x_1 + y_1, x_2 + y_2, \cdots, x_n + y_n)$

$\alpha \cdot (x_1, x_2, \cdots, x_n) = (\alpha x_1, \alpha x_2, \cdots, \alpha x_n)$

Show that, with these operations, $\mathbb{R}^n$ is a vector space.

c) [Optional] Consider the set

$\mathbb{W} = \{(x, y) : x \in \mathbb{R}, y \in \mathbb{R}, x > 0, y > 0\}$

On this set, define an addition operation and a scalar multiplication operation as follows:

$(x_1, y_1) + (x_2, y_2) = (x_1 y_1, x_2 y_2)$

$\alpha \cdot (x, y) = (x^\alpha, y^\alpha)$

Show that, with these operations, $\mathbb{W}$ is a vector space. Hint: Consider a bijection between $\mathbb{W}$ and the space $\mathbb{R}^2$ with $\mathbb{W} \ni (x, y) \mapsto (\log(x), \log(y)) \in \mathbb{R}^2$.

Given a normed space $(X, \|.\|)$, introduce a function:

$n(x, y) = \frac{\|x - y\|}{1 + \|x - y\|}$

Show that $n(x,y)$ satisfies the triangle inequality, $n(x,y) = 0$ iff $x = y$, and $n(x,y) = n(y,x)$.

A function which satisfies the above three properties is called a distance or a metric on $X$.

A non-empty subset $M$ of a (real) linear vector space $\mathbb{X}$ is called a subspace of $\mathbb{X}$ if

$\alpha x + \beta y \in M, \quad \forall x, y \in M \quad \text{and} \quad \alpha, \beta \in \mathbb{R}.$

In particular, the null element $\underline{0}$ is an element of every subspace.

For $M, N$ two subspaces of a vector space $\mathbb{X}$, show that $M \cap N$ is also a subspace of $\mathbb{X}$.

Let $1 \le p, q \le \infty$ with $1/p + 1/q = 1$. Let $x \in l_p(\mathbb{Z}_+)$ and $y \in l_q(\mathbb{Z}_+)$. Then,

$\sum_{i=0}^{\infty} |x_i y_i| \le \|x\|_p \|y\|_q$

This is known as Holder's inequality. Equality holds if and only if

$\left(\frac{x_i}{\|x\|_p}\right)^{(1/q)} = \left(\frac{y_i}{\|y\|_q}\right)^{(1/p)},$

for each $i \in \mathbb{Z}_+$.

To prove this, perform the following:

a) Show that for $a \ge 0, b \ge 0, c \in (0,1)$: $a^c b^{1-c} \le ca + (1-c)b$ with equality if and only if $a = b$. To show this, you may consider the function $f(t) = t^c - ct + c - 1$ and see how it behaves for $t \ge 0$ and let $t = a/b$.

b) Apply the inequality $a^c b^{1-c} \le ca + (1-c)b$ to the numbers:

$a = \left(\frac{|x_i|}{\|x\|_p}\right)^p, \quad b = \left(\frac{|y_i|}{\|y\|_q}\right)^q, \quad c = 1/p$

Holder's inequality is useful to prove Minkowski's inequality which states that for $1 < p < \infty$,

$\|x + y\|_p \le \|x\|_p + \|y\|_p$

a) Let $C([0,1]; \mathbb{R})$ be the space of continuous functions in $\Gamma([0,1]; \mathbb{R})$ with the norm

$\|f\| = \sup_{t \in [0,1]} |f(t)|.$

Is this space a complete normed linear space?

b) In class we will show that under the norm $\|f\| = \int_0^1 |f(t)| dt$, the space of continuous functions $C([0,1]; \mathbb{R})$ is not complete. Let us revisit this property.

Consider the sequence

$x_n(t) = \begin{cases} 1, & \text{if } 0 \le t \le 1/2 \\ -2^n(t - 1/2) + 1 & \text{if } 1/2 < t < (1/2) + (1/2)^n \\ 0, & \text{if } (1/2) + (1/2)^n \le t \le 1 \end{cases}$

Is this sequence Cauchy under the described norm? Does the sequence have a limit which is continuous?

Find a function $f \in L_2([0,2]; \mathbb{R})$ of unit norm, that is with

$\int_0^2 |f(t)|^2 dt = 1,$

which

a) maximizes

$J(f) = \int_0^2 f(t)\sin(\pi t) dt$

b) minimizes

$J(f) = \int_0^2 f(t)\sin(\pi t) dt$

Are the solutions unique?

Hint: Apply the Cauchy-Schwarz inequality.

Let $X$ be a pre-Hilbert space with norm $\|x\| = \sqrt{\langle x, x \rangle}$. Prove the Parallelogram law, that is, show that:

$\|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2$

Let $T : L_2(\mathbb{R}_+; \mathbb{R}) \to \mathbb{R}$ be a mapping given by:

$T(f) = \int_0^\infty f(t) \frac{1}{1 + t^2} dt$

Is $T$ continuous at some $f_0 \in L_2(\mathbb{R}_+; \mathbb{R})$?