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Section 3.1 basics of normed spaces

The basic idea of a normed space is: a vector space (where linearity makes sense), equipped with a compatible norm (a notion of size).

The fundamental example of a normed space is Euclidean space, the set of tuples (of a fixed length) of real numbers:

\begin{equation*} \mathbb{R}^m=\left\{\left(x_1,x_2,\ldots,x_m\right)\middle\vert x_1,x_2,\ldots, x_m\in \mathbb{R}\right\} \end{equation*}

In Euclidean space, the norm is

\begin{equation*} \left\lVert\left(x_1,x_2,\ldots,x_m\right)\right\rVert=\sqrt{x_1^2+x_2^2+\cdots+x_m^2} \end{equation*}

A normed space is supposed to be a generalization of this concept, in the same way that ordered fields are a generalization of the rational numbers.

Subsection 3.1.1 vector spaces

Definition 3.1.1.

A vector space with coefficients in \(\mathbb{R}\) is a set \(V\) together with operations

addition \(+\)
which takes two elements of \(V\) and results in an element of \(V\text{,}\)and
scaling \(\cdot\)
which takes a real number and an element of \(V\) and results in an element of \(V\text{.}\)

These operations satisfy the following axioms:

AC
For any \(x,y\in V, x+y=y+x\text{.}\)
AA
For any \(x,y,z\in V, (x+y)+z=x+(y+z)\text{.}\)
Z
There is \(\mathbf{0}\in V\) so that for any \(x\in V\text{,}\) \(\mathbf{0}+x=x\text{.}\)
AIn
Every \(x\in V\) has some \(-x\in V\) so that \(x+(-x)=\mathbf{0}\text{.}\)
SA
For any \(\lambda,\mu\in \mathbb{R},x\in V\text{,}\) \(\lambda\cdot (\mu \cdot x)=(\lambda\mu)\cdot x\text{.}\)
SDVA
For any \(\lambda\in\mathbb{R}, x,y\in V\text{,}\) \(\lambda\cdot(x+y)=(\lambda \cdot x)+(\lambda\cdot y)\text{.}\)
SDSA
For any \(\lambda,\mu\in \mathbb{R}, x\in V\text{,}\) \((\lambda+\mu)\cdot x=(\lambda\cdot x)+(\mu\cdot x)\text{.}\)
1
For any \(x\in V\text{,}\) \(1\cdot x=x\text{.}\)

Remark 3.1.2.

Notice that these axioms have a lot of overlap with the field axioms. Because of that, many of the same results (like "additive inverses are unique") hold in a vector space, using exactly the same proof.

Show that \(\mathbb{R}\) is a vector space with coefficients in \(\mathbb{R}\text{.}\)

Remark 3.1.4.

Checkpoint 3.1.3 tells us that, anytime we prove anything about a vector space, we've proved it about \(\mathbb{R}\text{.}\)

Example 3.1.5.

\(\mathbb{R}^m\) is a vector space with coefficents in \(\mathbb{R}\text{,}\) if we use the operations

\begin{equation*} \begin{cases} (x_1,\ldots,x_m)+(y_1,\ldots,y_m)=(x_1+y_1,\ldots,x_m+y_m)\\ \lambda\cdot (x_1,\ldots, x_m)=(\lambda x_1,\ldots, \lambda x_m)\end{cases} \end{equation*}

Example 3.1.6.

The set \(\mathcal{F}\) of all functions \(\mathbb{R}\to\mathbb{R}\) is a vector space with coefficients in \(\mathbb{R}\text{,}\) if we use the operations

\begin{equation*} \begin{cases} (f+g)(t)=f(t)+g(t)\\ (\lambda\cdot f)(t)=\lambda f(t)\end{cases} \end{equation*}

Properly speaking, the study of vector spaces belongs in a linear algebra class. We won't go into serious detail here because we're about other business. But you should learn linear algebra, and a lot of it.

Subsection 3.1.2 linear transformations

Given two vector spaces (call them \(V\) and \(W\)), we can ask which functions \(F:V\to W\) are mutually compatible with the operations on \(V\) and \(W\text{.}\) That is, we want \(F\) to satisfy:

Definition 3.1.7.

Given two vector spaces \((V,+_V,\cdot_V)\text{,}\) \((W,+_W,\cdot_W)\text{,}\) the function \(F:V\to W\) is linear if

  1. for any \(v_1,v_2\in V\text{,}\) \(F(v_1+_Vv_2)=F(v_1)+_WF(v_2)\text{,}\) and

  2. for any \(v\in V\text{,}\) \(\lambda\in \mathbb{R}\text{,}\) \(F(\lambda\cdot_Vv)=\lambda\cdot_W F(v)\text{.}\)

Remark 3.1.8.

The words function, transformation, and map will be used synonymously in these notes, but with a stylistic preference: I will habitually write linear transformation or linear map, while keeping the word function for any assignment of outputs to inputs.

If we take \(V=\mathbb{R}\) and \(W=\mathbb{R}\text{,}\) equipped with the ordinary operations you know and love. Suppose \(F:\mathbb{R}\to\mathbb{R}\) is a linear transformation. Define \(a=F(1)\text{.}\)

  1. What is \(F(n)\) for \(n\in\mathbb{N}\text{?}\)

  2. What is \(F(x)\) for \(x\in\mathbb{R}\text{?}\)

Put this together: you've just established that a linear transformation \(\mathbb{R}\to\mathbb{R}\) must look like. . . what?

Hint.

The answer may surprise you!

It's not quite true that the theory of linear transformations is the same as the study of matrices, but it's close enough to being true for our purposes.

Subsection 3.1.3 norms

If you look back up at the ideas we used in Chapter 1 and Chapter 2, the absolute value function plays a big role. Informally, \(\lvert r\rvert\) tells us how big \(r\in\mathbb{R}\) is. To define convergence of limits, we need to ask how big \(x_n-L\) is. At the very least, we'll want to have that same kind of functionality.

Definition 3.1.11.

A norm on a vector space \(V\) is a function \(\lVert\cdot\rVert:V\to \mathbb{R}\) which satisfies:

PD
For any \(x\in V\text{,}\) \(\lVert x\rVert\geq 0\text{;}\) the only way to get \(\lVert x\rVert=0\) is if \(x=\mathbf{0}\text{.}\)
T
For any \(x,y\in V\text{,}\) \(\lVert x+y\rVert\leq \lVert x\rVert+\lVert y\rVert\)
NS
For any \(x\in V, \lambda\in \mathbb{R}\text{,}\) \(\lVert \lambda\cdot x\rVert=\lvert \lambda\rvert \lVert x\rVert\)

Example 3.1.12.

For any \(1\leq p\lt\infty\text{,}\) we can define the \(p\)-norm on \(\mathbb{R}^m\) by

\begin{equation*} \left\lVert\left(x_1,\ldots,x_m\right)\right\rVert_p=\sqrt[p]{\lvert x_1\rvert^p+\cdots+\lvert x_m\rvert^p} \end{equation*}

Definition 3.1.13.

The closed unit ball in a normed space is the set of vectors \(x\) with \(\lVert x\rVert\leq 1\text{.}\)

Figure 3.1.14.

The closed unit ball in \(\mathbb{R}^2\text{,}\) equipped with the norm \(\lVert\cdot\rVert_p\text{.}\) Use the slider to change the value of \(p\) and see the effect on the unit ball.

Example 3.1.15.

The sup norm or \(\infty\)-norm on \(\mathbb{R}^m\) is

\begin{equation*} \left\lVert (x_1,\ldots,x_m)\right\rVert_{\infty}=\max\{\lvert x_1\rvert,\ldots,\lvert x_m\rvert\}\ \ . \end{equation*}

Prove that the sup norm on \(\mathbb{R}^2\) is a norm.

Sketch the unit ball, with respect to the sup norm on \(\mathbb{R}^2\text{.}\)

Example 3.1.18.

Set

\begin{equation*} \mathbb{R}^\infty=\left\{ \left(x_1,x_2,\ldots,x_k,\ldots\right)\middle\vert x_i\in \mathbb{R}\text{ and only finitely many }x_i\neq 0\right\} \end{equation*}

This is a vector space with componentwise addition and scaling.

For any \(p\in[1,\infty)\text{,}\) we can define the \(p\)-norm

\begin{equation*} \lVert (x_1,x_2,\ldots, x_k,\ldots)\rVert_p=\sqrt[p]{x_1^p+x_2^p+\cdots+x_k^p+\cdots} \end{equation*}

Subsection 3.1.4 Sequences in Normed Spaces

The notation \(\lVert\cdot\rVert\) is reminiscent of absolute value, and this is a conscious choice. It allows us to transfer most notions from an ordered field into a normed space. (But not, of course, the order part or the multiplicative inversion part -- then we'd just land back in an ordered field.)

By way of example:

Definition 3.1.19.

A sequence in the normed space \((V,\lVert\cdot\rVert)\) is a function \(X:\mathbb{N}\to V\text{.}\)

Definition 3.1.20.

We say that the sequence \(X\) in \((V,\lVert\cdot\rVert)\) converges to \(x\in V\) if

\begin{equation*} \forall \epsilon\gt 0, \exists K\in\mathbb{N}: n\gt K\Rightarrow \lVert x_n-x\rVert\lt \epsilon \end{equation*}

Which of the results in Section 2.2 apply in a normed space? For any that do, formulate and prove the statement.

Which of the definitions in Section 2.3 and Section 2.4 apply in a normed space? For any that do, write the definition out.

Which of the results in Section 2.3 and Section 2.4 apply in a normed space? For any that do, formulate and prove the statement.