---
title: Orthogonality
parent: Inner Product Spaces
grand_parent: Functional Analysis Basics
nav_order: 1
---

# {{ page.title }}

{% definition Orthogonal Vectors %}
Two vectors $x$ and $y$ of an inner product space $(X,\innerp{\cdot}{\cdot})$
are said to be *orthogonal* or *perpendicular* if $\innerp{x}{y} = 0$,
and this fact is indicated by writing $x \perp y$. \
A set $A \subset X$ is called orthogonal,
if the elements of $S$ are pairwise orthogonal to each other.
{% enddefinition %}

{% theorem * Pythagoras’ Theorem %}
If $x$ and $y$ are orthogonal vectors of an inner product space, then

$$
\norm{x+y}^2 = \norm{x}^2 + \norm{y}^2.
$$

More generally,if $\braces{x_1,\ldots,x_n}$ is an orthogonal set in an inner product space, then

$$
\norm{x_1 + \cdots + x_n}^2 =
\norm{x_1}^2 + \cdots + \norm{x_n}^2.
$$
{% endtheorem %}

The converse of Pythagoras’ Theorem is true for real inner product space,
but false in the complex case.
For example, let $x$ be any unit vector in a complex inner product space.
Then $x$ is not orthogonal to $ix$, since $\innerp{x}{ix} = i \ne 0$.
However, $\norm{x+ix}^2 = \abs{1+i}^2 = 2 = 1 + 1 = \norm{x}^2 + \norm{ix}^2$.

{% definition Orthonormal Set %}
A subset $S$ of an inner product space is called *orthonormal*
if we have for all $x,y \in S$

$$
\innerp{x}{y} = \begin{cases}
0 & x=y, \\
1 & x \ne y.
\end{cases}
$$
{% enddefinition %}

In other words, an orthonormal set is an orthogonal set of unit vectors.

{% proposition %}
Every orthonormal set is linearly independent.
{% endproposition %}

{% proof %}
Suppose that $\braces{x_1,\ldots,x_n}$ is a finite subset of $S$ and that

$$
\alpha_1 x_1 + \cdots + \alpha_n x_n = 0
$$

for some scalars $\alpha_1,\ldots,\alpha_n$.
Application of $\innerp{x_i}{\cdot}$ yields
$\alpha_i = 0$ for all $i \in \braces{1,\ldots,n}$.
{% endproof %}

Recall that a subset $S$ of a normed space $X$ is called total
if its span is dense in $X$.

{% definition Orthonormal Basis %}
A total orthonormal set in an inner product space is called
*orthonormal basis* (or *complete orthonormal system*).
{% enddefinition %}

{% theorem %}
Every Hilbert space has an orthonormal basis.
{% endtheorem %}