Inner product space

An inner product space is a vector space with an inner product.

An inner product is an operation that assigns a scalar to a pair of vectors \langle\boldsymbol{\mathit{u}}\vert\boldsymbol{\mathit{v}}\rangle or a scalar to a pair of functions \langle f\vert g\rangle. The way to assign the scalar may be through the matrix multiplication of the pair of vectors, e.g.

\langle\boldsymbol{\mathit{u}}\vert\boldsymbol{\mathit{v}}\rangle=\begin{pmatrix} u_{1}^{*} &u_{2}^{*}& \cdots &u_{N}^{*} \end{pmatrix}\begin{pmatrix} v_1\\v_2 \\ \vdots \\ v_N \end{pmatrix}=\sum_{i=1}^{N}u_{i}^{*}v_i\; \; \; \; \; \; \; \; 3

or it may be through an integral of the pair of functions.

\langle f\vert g\rangle=\int_{-\infty}^{\infty}f(x)^{*}g(x)dx

You may notice that eq3 resembles a dot product. The dot product pertains to vectors in \mathbb{R}^{3}, where \boldsymbol{\mathit{A}}\cdot\boldsymbol{\mathit{B}}=\sum_{i=1}^{3}A_iB_i, which can be extended to N-dimensions, where \langle\boldsymbol{\mathit{A}}\vert\boldsymbol{\mathit{B}}\rangle=\sum_{i=1}^{N}A_iB_i, and to include complex and real functions, \langle f\vert g\rangle=\int_{-\infty}^{\infty}f(x)^{*}g(x)dx. Therefore, an inner product is a generalisation of the dot product.

An inner product space has the following properties:

  1. Conjugate symmetry: \langle f\vert g\rangle=\langle g\vert f\rangle^{*}
  2. Additivity: \langle f+g\vert h\rangle=\langle f\vert h\rangle+\langle g\vert h\rangle
  3. Positive semi-definiteness: \langle f\vert f\rangle\geq 0, with \langle f\vert f\rangle= 0 if f=0



i) Why is the inner product space positive semi-definite? ii) Show that orthogonal vectors are linearly independent.


i) A general vector space of \langle\boldsymbol{\mathit{a}}\vert\boldsymbol{\mathit{a}}\rangle can be positive or negative. The inner product space is defined such that \langle f\vert f\rangle\geq 0, with \langle f\vert f\rangle= 0 if f=0, which is useful in quantum mechanics.

ii) Let the set of vectors \left \{ \boldsymbol{\mathit{v_k}} \right \} in eq1 be orthogonal vectors. The dot product of eq1 with \boldsymbol{\mathit{v_i}} gives c_i\boldsymbol{\mathit{v_i}}\cdot\boldsymbol{\mathit{v_i}} =c_i\left |\boldsymbol{\mathit{v_i}} \right |^{2}=0. Since the magnitudes of orthogonal vectors are non-zero, c_i=0. Hence, orthogonal vectors are linearly independent.


Two functions (or two vectors) are orthogonal if \langle f\vert g\rangle= 0. Elements of a set of basis functions are orthonormal if \langle \phi_i\vert \phi_j\rangle=\delta_{ij} where

\delta_{ij}= \{\; \begin{matrix} 1 & for\; \; i=j\\ 0 & for\; \; i\neq j \end{matrix}

In other words, two functions (or two vectors) are orthonormal if they are orthogonal and normalised.

Finally, the norm (or length) of a vector \boldsymbol{\mathit{u}} is denoted by \left \|\boldsymbol{\mathit{u}}\right \| and is defined as \left \|\boldsymbol{\mathit{u}}\right \|=\sqrt{\langle\boldsymbol{\mathit{u}}\vert\boldsymbol{\mathit{u}}\rangle}=\sqrt{\left |\boldsymbol{\mathit{u}}\right |\left |\boldsymbol{\mathit{u}}\right |cos\: 0^{\circ} }=\left |\boldsymbol{\mathit{u}}\right |. With this association of inner product and the length of a vector, we can establish the relationship between inner product \langle\boldsymbol{\mathit{u}}\vert\boldsymbol{\mathit{v}}\rangle and the Euclidean distance d(\boldsymbol{\mathit{u}},\boldsymbol{\mathit{v}}) between 2 vectors \boldsymbol{\mathit{u}} and \boldsymbol{\mathit{v}}. Using the \mathbb{R}^{2} space as an example, where \boldsymbol{\mathit{u}}=\begin{pmatrix} 6\\9 \end{pmatrix} and \boldsymbol{\mathit{v}}=\begin{pmatrix} 3\\5 \end{pmatrix}, we have

d(\boldsymbol{\mathit{u}},\boldsymbol{\mathit{v}})=\left \|\boldsymbol{\mathit{u}}-\boldsymbol{\mathit{v}}\right \|=\sqrt{\langle\boldsymbol{\mathit{u}}- \boldsymbol{\mathit{v}}\vert\boldsymbol{\mathit{u}}- \boldsymbol{\mathit{v}}\rangle}=\sqrt{\langle\begin{pmatrix} 3\\4 \end{pmatrix}\vert \begin{pmatrix} 3\\4 \end{pmatrix}\rangle}

=\sqrt{\begin{pmatrix} 3 &4 \end{pmatrix}\begin{pmatrix} 3\\4 \end{pmatrix}}=5



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