Inner Product Spaces Review

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 $<u, v>$ 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
$<u, u> ≥ 0$ $<u, u> = 0$ if and only $u = 0$. $<u + v, w> = <u, w> + <v, w>$ $<au, v> = a<u, v>$ $<u, v> = \overline{<v, u>}$
  • We also saw that inner product spaces also had additivity in the second slot, that is $<u, v + w> = <u, v> + <u, w>$ for all $u, v, w \in V$ and conjugate homogeneity in the second slot, that is $<u, av> = \overline{a}<u, v>$ 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 $<u, v> = u_1v_1 + u_2v_2 + ... + u_nv_n$. Another example of an inner product space is defined on $\wp (\mathbb{R})$ as $<p(x), q(x)> = \int_0^1 p(x)q(x) \: dx$ 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 $<u, v> = 0$ for $u, v \in V$ then $u$ and $v$ are said to be Orthogonal to each other.
(1)
\begin{align} \quad <u, v> = \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 <u, v> = \frac{ \| u + v \|^2 - \| u - v \|^2 + \| u + iv \|^2i - \| u - iv \|^2 i}{4} \end{align}
(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 <u, v> \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}
(6)
\begin{align} \quad \| u + v \|^2 + \| u - v \|^2 = 2 \| u \|^2 + 2 \| v \|^2 \end{align}
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