Lecture 04 - Subspaces

#math

Read up on Spectral Theory
Dym - Applied Functional Analysis

Complete Subspaces

We start with the proposition we ended with in the last lecture.

Proposition

i) Any closed subspace of a Banach/Hilbert space is complete, hence is also a Banach/Hilbert space.
ii) Any complete subspace is closed.
iii) The closure of a subspace is a subspace.

Proof

i) Since the subspace is closed, then every convergent sequence converges inside the set thus is complete.
ii) Since every Cauchy sequence in convergent and is contained in the subspace (and every convergent sequence is Cauchy) thus our space must be closed.
iii) Let $u, v \in \overset{―}{S}$ , assume that $u, v \notin S$ (that is $u, v$ is a limit point and not in the original subspace) then $\exists u_{n}, v_{n} \in S$ whose limit is $u, v$ respectively. Then, $α u_{n} + β v_{n} \in S$ and notice that:

| | (α x_{n} + β y_{n}) - (α x + β y) | | = | | α (x_{n} - x) + β (y_{n} - y) | | \leq | α | | | x_{n} - x | | + | β | | | y_{n} - y | |

that is it has the limit $α u + β v \in \overset{―}{S}$ that is $\overset{―}{S}$ is closed under scalar multiplication and vector addition, that is it indeed is a linear subspace.

Examples of Incomplete subspaces

Definition

We would define $C_{00}$ to be the space of sequences finite length that is $\exists N$ with $x_{n} = 0$ for $n \geq N$

It is clear that $C_{00} \subset l_{2}$ but notice that if we take the sequence (of finite sequences)

x_{n} = (1, 1 / 2, . . ., 1 / n, 0, 0, 0, . . .)

then $x_{n} \in C_{00}$ however the limit

x = (1, 1 / 2, . . ., 1 / n, . . .)

is not. Thus $C_{00}$ is not closed (and thus is not complete). However if we take the subspace:

S : {(a, 0, 0, 0, . . .) | a \in R}

Then $S$ is closed (and thus complete) in an incomplete space $C_{00}$ (that is in a complete space $l_{2}$ ).

Another example to consider is to take the subspace $C [0, 1] \subset L^{2} [0, 1]$ and consider the following sequence of functions:
$Pasted image 20240904103359.png$

It is continuous in $[0, 1]$ however the limit
$Pasted image 20240904104241.png$
Is not continuous. Thus $C [0, 1]$ is another example of a space that is incomplete.

Definition

We define a Hilbert space $L^{2} ([a, b])$ to be the smallest complete inner product space containing $C ([a, b])$ , with the inner product defined to be:

⟨ f (t), g (t) ⟩ = \int_{a}^{b} f (t) g (t) d t