\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
N-Dimensional Space
Real Norm
Space Inner Product
Needed by:
Complex Inner Product
Orthonormal Set of Real Vectors
Probability Vectors
Real Halfspaces
Real Inner Products
Real Polar Sets
Vectors as Matrices
Sheet PDF
Graph PDF

Real Inner Product


The real inner product (or dot product, scalar product) of two real vectors $x, y \in \R ^n$ is

\[ x_1y_1 + x_2y_2 + \cdots + x_ny_n \]

We denote the inner product of $x$ and $y$ by $\ip{x,y}$.


The inner product has several important properties

  1. $\ip{\alpha x, y} = \alpha\ip{x, y}$
  2. $\ip{x + y, z} = \ip{x, z} + \ip{y, z}$
  3. $\ip{x, y} = \ip{y, x}$
  4. $\ip{x,x} \geq 0$
  5. $\ip{x,x} = 0 \iff x = 0$

Connection to norm

It is important to note that $\norm{x} = \sqrt{\ip{x,x}}$.

Copyright © 2023 The Bourbaki Authors — All rights reserved — Version 13a6779cc About Show the old page view