# Inner Product Spaces Review

We will now review some of the recent content regarding inner product spaces.

- Recall from the Inner Product Spaces page that if $V$ is a vector space over $\mathbb{R}$ or $\mathbb{C}$ then an
**Inner Product**on $V$ is a function which takes every pair of vectors $u, v \in V$ and maps it to a number $__$__that satisfies the following properties for all $u, v, w \in V$ and $a \in \mathbb{F}$ ($\mathbb{R}$ or $\mathbb{C}$):

Positivity Property |
Definiteness Property |
Additivity in First Slot |
Homogeneity in First Slot |
Conjugate Symmetry Property |
---|---|---|---|---|

$ ≥ 0$ |
$ = 0$ if and only $u = 0$. |
$ = + |
$$ |
$ = \overline{ |

- We also saw that inner product spaces also had additivity in the second slot, that is $
__=__for all $u, v, w \in V$ and conjugate homogeneity in the second slot, that is $__+____$____= \overline{a}__for all $u, v \in V$ and $a \in \mathbb{F}$ ( $\mathbb{R}$ or $\mathbb{C}$).__$__

- Perhaps the simplest inner product space in the generic dot product defined on $\mathbb{R}^n$. For $u = (u_1, u_2, …, u_n)$ and $v = (v_1, v_2, …, v_n)$ we have that $
__= u_1v_1 + u_2v_2 + … + u_nv_n$__. Another example of an inner product space is defined on $\wp (\mathbb{R})$ as $= \int_0^1 p(x)q(x) \: dx$

for all $p(x), q(x) \in \wp(\mathbb{R})$.

- If $V$ is a vector space with an inner product then $V$ is called an
**Inner Product Space**.

- Furthermore if $
__= 0$__for $u, v \in V$ then $u$ and $v$ are said to be**Orthogonal**to each other.

- On the Formulas for The Inner Product we proved the following formulas for inner products. If $V$ is an inner product space over $\mathbb{R}$ then:

(1)

\begin{align} \quad

__= \frac{ \| u + v \|^2 – \| u – v \|^2}{4} \end{align}__- If $V$ is an inner product space over $\mathbb{C}$ then:

(2)

\begin{align} \quad

__= \frac{ \| u + v \|^2 – \| u – v \|^2 + \| u + iv \|^2i – \| u – iv \|^2 i}{4} \end{align}__- On The Pythagorean Theorem for Inner Product Spaces we proved the famous
**Pythagorean Theorem**(for inner product spaces) which says that if $V$ is an inner product space and if $u, v \in V$ are orthogonal then:

(3)

\begin{align} \quad \| u + v \|^2 = \| u \|^2 + \| v \|^2 \end{align}

- On The Cauchy-Schwarz Inequality page we proved one of the most famous inequalities in mathematics known as the
**Cauchy-Schwarz Inequality**which says that if $V$ is an inner product space then for all $u, v \in V$ with equality holding if and only if $u$ is a multiple of $v$:

(4)

\begin{align} \quad \mid

__\mid ≤ \| u \| \| v \| \end{align}__- We then proved the
**Triangle Inequality**on The Triangle Inequality for Inner Product Spaces page which says that if $V$ is an inner product space then for all $u, v \in V$ with equality holding if and only if $u$ is a nonnegative scalar multiple of $v$, we have that:

(5)

\begin{align} \quad \| u + v \| ≤ \| u \| + \| v \| \end{align}

- Then we proved
**The Parallelogram Identity**on The Parallelogram Identity for Inner Product Spaces page which says that if $u, v \in V$ then:

(6)

\begin{align} \quad \| u + v \|^2 + \| u – v \|^2 = 2 \| u \|^2 + 2 \| v \|^2 \end{align}

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