Inner Products and Inner Product Spaces Review]

We will now review some of the recent material regarding inner products and inner product spaces.

• On the Inner Products and Inner Product Spaces page we said that an Inner Product on a linear space $X$ is a function $\langle \cdot, \cdot \rangle : X \to \mathbb{C}$ that satisfies the following properties for all $x, y, z \in X$ and for all $\lambda \in \mathbb{C}$:

(1)

\begin{align} \quad \langle x, x \rangle & \geq 0 \quad \mathrm{and} \quad \langle x, x \rangle = 0 \: \mathrm{if \:and \: only \: if} \: x = 0 \\ \quad \langle x, y \rangle & = \overline{\langle y, x \rangle} \\ \quad \langle x + y, z \rangle & = \langle x, z \rangle + \langle y, z \rangle \\ \quad \langle \lambda x, y \rangle & = \lambda \langle x, y \rangle \end{align}

• We then proved another important set of properties for inner products. We proved that for all $x, y, z \in X$ and for all $\lambda \in \mathbb{C}$ that:

(2)

\begin{align} \quad \langle x, y + z \rangle &= \langle x, y \rangle + \langle x, z \rangle \\ \quad \langle x, \lambda y \rangle &= \overline{\lambda} \langle x, y \rangle \end{align}

• We then said that an Inner Product Space is a linear space equipped with an inner product.

• On The Cauchy-Schwarz Inequality for Inner Product Spaces page we proved the very important Cauchy-Schwarz inequality for inner product spaces which says that if $H$ is an inner product space then for all $x, y \in H$ we have that:

(3)

\begin{align} \quad | \langle x, y \rangle | \leq \langle x, x \rangle^{1/2} \langle y, y \rangle^{1/2} \end{align}

• On The Normed Space Induced by an Inner Product page we proved that if $H$ is an inner product space then the function $\| \cdot \| : H \to [0, \infty)$ defined below is a norm on $H$ and is called the Norm Induced by the Inner Product on $H$:

(4)

\begin{align} \quad \| x \| = \langle x, x \rangle^{1/2} \end{align}

• On The Parallelogram Identity for the Norm Induced by an Inner Product page we proved the parallelogram identity for the norm induced by an inner product, which says that for all $x, y \in H$ we have that:

(5)

\begin{align} \quad \| x + y \|^2 + \| x - y \|^2 = 2 \| x \|^2 + 2 \| y \|^2 \end{align}

• On the Orthogonal Sets in an Inner Product Space page we said that if $H$ is an inner product space then two elements $x, y \in H$ are said to be Orthogonal written $x \perp y$ if:

(6)

\begin{align} \quad \langle x, y \rangle = 0 \end{align}

• Furthermore, $x$ is said to be Orthogonal to the set $S \subseteq H$ written $x \perp S$ if for all $y \in S$ we have that:

(7)

\begin{align} \quad \langle x, y \rangle = 0 \end{align}

• We then said that if $S \subseteq H$ then the Orthogonal Set of $S$ denoted by $S^{\perp}$ is the set of all elements in $H$ that are orthogonal to $S$, that is:

(8)

\begin{align} \quad S^{\perp} = \{ x \in H : x \perp S \} \end{align}