Inner Product Space

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An Inner Product Space is a Vector Space for which an inner product is defined and the following properties hold. An finite dimensional Inner Product Space is called a Pre-Hilbert Space. For $\mathbf{u},\mathbf{v},\mathbf{w} \in V$ (V is eather a real or a complex Vector Space) and $\lambda \in \mathbb{C}$

  • IPS(1) $\hspace{1cm}$ $ \langle \mathbf{v}+\mathbf{v}' , \mathbf{w} \rangle =\langle \mathbf{v} , \mathbf{w} \rangle +\langle \mathbf{v}' , \mathbf{w} \rangle$ and $ \langle \mathbf{v} ,\lambda \mathbf{w} \rangle =\lambda \langle \mathbf{v} , \mathbf{w} \rangle $ (linear in the second argument)

  • IPS(2) $\hspace{1cm}$ $ \langle \mathbf{v} , \mathbf{w} \rangle = \overline{\langle \mathbf{w} , \mathbf{v} \rangle }$ (hermitian)

  • IPS(3) $\hspace{1cm}$ $ \langle \mathbf{v} , \mathbf{v} \rangle \geq 0$ and $\langle \mathbf{v} , \mathbf{v} \rangle=0$ iff $\mathbf{v}=0$ (positive definite)

Moreover, from IPS(1) and IPS(2) it follows that the inner product (also called scalar product) is sesquilinear in the first argument

  • $ \langle \lambda \mathbf{v} , \mathbf{w} \rangle =\lambda^* \langle \mathbf{v} , \mathbf{w} \rangle $

Real Inner Product

The inner product of a real Vector Space $V$ is known as dot product. We define the dot product of two vectors $\mathbf{v},\mathbf{w} \in V$ as

\begin{align} (\mathbf{v},\mathbf{w}):=\sum_{i=1}^n v_i w_i \tag{1} \end{align}

There exist several ways of writing the dot product, they are equivalent \begin{align} (\mathbf{v},\mathbf{w})= \mathbf{v} \cdot \mathbf{w} =\mathbf{v}^T \mathbf{w} \end{align}

Complex Inner Product

if $\mathbf{v},\mathbf{w} \in W$ are elements of a complex Vector Space $W$ we define the inner product

\begin{align} \langle \mathbf{v},\mathbf{w} \rangle :=\sum_{i=1}^n v_i^* w_i \tag{2} \end{align}

$v_i$ and $w_i$ are complex numbers $v_i,w_i \in \mathbb{C}$ and $^*$ stands for the complex conjugate. In fact (1) is just a special case of (2) since $\mathbb{R} \subset \mathbb{C}$ and for $v_i \in \mathbb{R}$ we simply get $v^*_i=v_i$.

Continuous Inner Product

We can even define a product of two functions. To motivate this desire, let us study the solution of the second order differential equation \begin{align} x''+\omega^2 x =0 \end{align} the set of linear independent solutions is \begin{align} \{x_1(t), x_2(t) \}=\{ e^{-i \omega t}, e^{i \omega t} \} \end{align} we can regard these solution as a basis of a Vector Space $X$. Every linear combination $a e^{-i \omega t} + b e^{i \omega t}$ is an element of this vector space, it is the analog of a vector in euclidian space. Often the basis \begin{align} \{sin(\omega t), cos(\omega t) \}=\{ \frac{e^{i \omega t}-e^{-i \omega t}}{2i} , \frac{e^{i \omega t}+e^{-i \omega t}}{2} \} \end{align} where sine and cosine are just linear combinations of $\{ e^{-i \omega t}, e^{i \omega t} \}$. Notice, in a linear vector space, the sum of vectors in a vector space is a again a vector of this vector space. This means every function \begin{align} f(t)=\sum_{n} a_n cos(nt) + b_n sin(nt) \tag{3} \end{align} (a Fourier Series) is again a solution of the differential equation. The field that generalizes the concept of linear algebra to functions is called Functional Analysis. In the definition of the Fourier Series I discussed the orthogonality of sine and cosine. We arrived at the expressions \begin{align} \int_{-\pi}^{\pi} cos(kx) \, cos(lx)dx=\left\{\begin{array}{ll} 2\pi & \text{for } k=l=0 \\ \pi & \text{for } k=l>0 \\ 0 & \text{for } k\neq l=0 \end{array}\right. \tag{4} \end{align} \begin{align} \int_{-\pi}^{\pi} sin(kx) \, sin(lx)dx=\left\{\begin{array}{ll} 0 & \text{for } k=l=0 \\ \pi & \text{for } k=l>0 \\ 0 & \text{for } k\neq l=0 \end{array}\right. \tag{5} \end{align} \begin{align} \int_{-\pi}^{\pi} sin(kx) \, cos(lx)dx= 0 \hspace{0.9cm} \text{ for all } k \text{ und } l \tag{6} \end{align}

so why don't we define an inner product of the form

\begin{align} \langle f_n,g_m \rangle =\int_{a}^{b} f_n(x) \, g_m(x) \, dx \tag{7} \end{align}