Editing Cauchy–Schwarz inequality (section)

== Special cases ==

=== Sedrakyan's lemma – positive real numbers ===
[[Sedrakyan's inequality]], also known as [[Harald Bergström|Bergström]]'s inequality, [[Arthur Engel (mathematician)|Engel]]'s form, [[Titu Andreescu|Titu]]'s lemma (or the T2 lemma), states that for real numbers <math>u_1, u_2, \dots, u_n</math> and positive real numbers <math>v_1, v_2, \dots, v_n</math>:
<math display=block>\frac{\left(u_1 + u_2 + \cdots + u_n\right)^2}{v_1 + v_2 + \cdots + v_n} \leq \frac{u^2_1}{v_1} + \frac{u^2_2}{v_2} + \cdots + \frac{u^2_n}{v_n},</math>
or, using summation notation,
<math display=block>\biggl(\sum_{i=1}^n u_i\biggr)^2 \bigg/ \sum_{i=1}^n v_i \,\leq\, \sum_{i=1}^n \frac{u_i^2}{v_i}.</math>

It is a direct consequence of the Cauchy–Schwarz inequality, obtained by using the [[dot product]] on <math>\R^n</math> upon substituting <math>u_i' = \frac{u_i}{\sqrt{v_i\vphantom{t}}}</math> and <math>v_i' = {\textstyle \sqrt{v_i\vphantom{t}}}</math>. This form is especially helpful when the inequality involves fractions where the numerator is a [[Square number|perfect square]].

=== {{math|R<sup>2</sup>}} - The plane ===
[[File:Cauchy-Schwarz inequation in Euclidean plane.gif|thumb|300px|Cauchy–Schwarz inequality in a unit circle of the Euclidean plane]]
The real vector space <math>\R^2</math> denotes the 2-dimensional plane. It is also the 2-dimensional [[Euclidean space]] where the inner product is the [[dot product]]. 
If <math>\mathbf{u} = (u_1, u_2)</math> and <math>\mathbf{v} = (v_1, v_2)</math> then the Cauchy–Schwarz inequality becomes:
<math display=block>\langle \mathbf{u}, \mathbf{v} \rangle^2 = \bigl(\|\mathbf{u}\| \|\mathbf{v}\| \cos \theta\bigr)^2 \leq \|\mathbf{u}\|^2 \|\mathbf{v}\|^2,</math>
where <math>\theta</math> is the [[angle]] between <math>\mathbf{u}</math> and <math>\mathbf{v}</math>.

The form above is perhaps the easiest in which to understand the inequality, since the square of the cosine can be at most 1, which occurs when the vectors are in the same or opposite directions. It can also be restated in terms of the vector coordinates <math>u_1</math>, <math>u_2</math>, <math>v_1</math>, and <math>v_2</math> as
<math display=block>\left(u_1 v_1 + u_2 v_2\right)^2 \leq \left(u_1^2 + u_2^2\right) \left(v_1^2 + v_2^2\right),</math>
where equality holds if and only if the vector <math>\left(u_1, u_2\right)</math> is in the same or opposite direction as the vector <math>\left(v_1, v_2\right)</math>, or if one of them is the zero vector.

=== {{anchor|Real Euclidean space}}{{math|'''R'''<sup>''n''</sup>}}: ''n''-dimensional Euclidean space ===
{{anchor|real number proof}}In [[Euclidean space]] <math>\R^n</math> with the standard inner product, which is the [[dot product]], the Cauchy–Schwarz inequality becomes:
<math display=block>\biggl(\sum_{i=1}^n u_i v_i\biggr)^2 \leq \biggl(\sum_{i=1}^n u_i^2\biggr) \biggl(\sum_{i=1}^n v_i^2\biggr).</math>

The Cauchy–Schwarz inequality can be proved using only elementary algebra in this case by observing that the difference of the right and the left hand side is 
<math display=block> \tfrac{1}{2} \sum_{i=1}^n\sum_{j=1}^n (u_i v_j - u_j v_i)^2 \ge 0</math> 

or by considering the following [[quadratic polynomial]] in <math>x</math>
<math display=block> (u_1 x + v_1)^2 + \cdots + (u_n x + v_n)^2 = \biggl(\sum_i u_i^2\biggr) x^2 + 2 \biggl(\sum_i u_i v_i\biggr) x + \sum_i v_i^2.</math>

Since the latter polynomial is nonnegative, it has at most one real root, hence its [[discriminant]] is less than or equal to zero. That is,
<math display=block>\biggl(\sum_i u_i v_i\biggr)^2 - \biggl(\sum_i {u_i^2}\biggr)  \biggl(\sum_i {v_i^2}\biggr) \leq 0.</math>

=== {{math|'''C'''<sup>''n''</sup>}}: ''n''-dimensional complex space===
If <math>\mathbf{u}, \mathbf{v} \in \Complex^n</math> with <math>\mathbf{u} = (u_1, \ldots, u_n)</math> and <math>\mathbf{v} = (v_1, \ldots, v_n)</math> (where <math>u_1, \ldots, u_n \in \Complex</math> and <math>v_1, \ldots, v_n \in \Complex</math>) and if the inner product on the vector space <math>\Complex^n</math> is the canonical complex inner product (defined by <math>\langle \mathbf{u}, \mathbf{v} \rangle := u_1 \overline{v_1} + \cdots + u_{n} \overline{v_n},</math> where the bar notation is used for [[Complex conjugate|complex conjugation]]), then the inequality may be restated more explicitly as follows: 
<math display=block>\bigl|\langle \mathbf{u}, \mathbf{v} \rangle\bigr|^2 
= \Biggl|\sum_{k=1}^n u_k\bar{v}_k\Biggr|^2 
\leq \langle \mathbf{u}, \mathbf{u} \rangle \langle \mathbf{v}, \mathbf{v} \rangle 
= \biggl(\sum_{k=1}^n u_k \bar{u}_k\biggr) \biggl(\sum_{k=1}^n v_k \bar{v}_k\biggr) 
= \sum_{j=1}^n |u_j|^2 \sum_{k=1}^n |v_k|^2.</math>

That is,
<math display=block>\bigl|u_1 \bar{v}_1 + \cdots + u_n \bar{v}_n\bigr|^2 \leq \bigl(|u_1|{}^2 + \cdots + |u_n|{}^2\bigr) \bigl(|v_1|{}^2 + \cdots + |v_n|{}^2\bigr).</math>

=== {{math|''L''<sup>2</sup>}} ===

For the inner product space of [[square-integrable]] complex-valued [[function (mathematics)|functions]], the following inequality holds.
<math display=block>\left|\int_{\R^n} f(x) \overline{g(x)}\,dx\right|^2  \leq  \int_{\R^n} \bigl|f(x)\bigr|^2\,dx \int_{\R^n} \bigl|g(x)\bigr|^2 \,dx.</math>

The [[Hölder inequality]] is a generalization of this.