Measure Theory/L2 Inner Product Space

Inner Products

We have defined, in Lesson 0, the inner product for $L^{2}(E)$. However, when calling something by the name "inner product" we mean to communicate a few properties. We should ensure that our inner product,

$\langle f,g\rangle =\int _{E}fg$

deserves the name.

Besides justifying the name, out of a sense of principle, these properties will also be useful common manipulations in the proofs of the theorems that we care about -- the most important, for this section, being completeness.

Definition: inner product

Let V be any vector space over ${\mathbb {R}}$, and let $\langle \cdot ,\cdot \rangle :V\to {\mathbb {R}}$ be any binary function. In all of the definitions below, let ${\vec {v}},{\vec {w}},{\vec {x}}\in V$ be arbitrary vectors and $a\in {\mathbb {R}}$ an arbitrary real number.

We define $\langle \cdot ,\cdot \rangle$ to be symmetric if

$\langle {\vec {v}},{\vec {w}}\rangle =\langle {\vec {w}},{\vec {v}}\rangle$

We define it to be linear in the first coordinate (or just linear for short) if

$\langle a{\vec {v}}+{\vec {w}},{\vec {x}}\rangle =a\langle {\vec {v}},{\vec {x}}\rangle +\langle {\vec {w}},{\vec {x}}\rangle$

We define it to be positive definite if

$\langle {\vec {v}},{\vec {v}}\rangle =\|{\vec {v}}\|\geq 0$ with equality holding if and only if ${\vec {v}}={\vec {0}}$.

We define $\langle \cdot ,\cdot \rangle$ to be an inner product on V if it is symmetric, linear, and positive definite.

Recall that you have already decided what ${\vec {0}}$ is when the vector space in question is $L^{2}(E)$, in the previous lesson, Lesson 1.

Exercise 1. L2 Inner Product Is an Inner Product Show that the $L^{2}(E)$ inner product is symmetric and linear. (This should be trivial.) Also show that for every ${\vec {v}}\in V$, we have $\|{\vec {v}}\|_{2}\geq 0$. Also show that $\|{\vec {0}}\|_{2}=0$. Now consider proving that if $\|{\vec {v}}\|=0$ then ${\vec {v}}={\vec {0}}$. First show that, in fact, there is a function $f\in {\mathcal {L}}^{2}(E)$ for which $\int _{E}f^{2}=0$ even though $f\neq 0$. Next recall that, technically, $L^{2}(E)$ is a set of equivalence classes. Recalling also a result from a previous section, prove that if $\|{\vec {v}}\|_{2}=0$ then as an equality of equivalence classes, it follows that ${\vec {v}}={\vec {0}}$. Conclude that the $L^{2}(E)$ inner product, is an inner product on $L^{2}(E)$.

Normed Space

Definition: norm

Let V be a vector space and $\|\cdot \|:V\to {\mathbb {R}}$ a function. Let ${\vec {v}}\in V$ be an arbitrary vector and $a\in {\mathbb {R}}$ an arbitrary real number, throughout this definition. We say that $\|\cdot \|$ is positive definite if

$\|{\vec {v}}\|\geq 0$ with equality holding if and only if ${\vec {v}}={\vec {0}}$

We call it homogeneous if

$\|a{\vec {v}}\|=|a|\|{\vec {v}}\|$

We call it subadditive if

$\|{\vec {v}}+{\vec {w}}\|\leq \|{\vec {v}}\|+\|{\vec {w}}\|$

(We call the inequality itself, $\|{\vec {v}}+{\vec {w}}\|\leq \|{\vec {v}}\|+\|{\vec {w}}\|$, the triangle inequality.)

We call any function a norm if it is positive definite, homogeneous, and subadditive. We call V a normed vector space if it has a norm.

Exercise 2. L2 Is a Normed Metric Space Show that every inner product space is a normed space, and infer that $L^{2}(E)$ is a normed space with norm $\|{\vec {v}}\|_{2}=\langle {\vec {v}},{\vec {v}}\rangle$.

Metric Space

Definition: metric space

Let X be a set and $d:X\times X\to {\mathbb {R}}$. Let $v,w,x\in X$. We say that d is positive definite if

$d(v,w)\geq 0$ with equality if and only if $v=w$

We say that d is symmetric if

$d(v,w)=d(w,v)$

We call the inequality

$d(v,w)\leq d(v,x)+d(x,w)$

the triangle inequality. (If we wish, we may distinguish between the norm and distance "triangle inequality".)

The function d is called a distance metric if it is positive definite, symmetric, and satisfies the triangle inequality. The space X is called a metric space if it has a distance metric.

Exercise 3. L2 Is a Metric Space Prove that every normed space is a metric space, and infer that $L^{2}(E)$ is a metric space with metric $d({\vec {v}},{\vec {w}})=\|{\vec {v}}-{\vec {w}}\|_{2}$.