Inner Product

Topics: Algebra - Vector Space - Calculus

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(definition)

Let $V$ be a vector space on $\mathbb{K}$ and let $⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{K}$ be a function.

We say that $⟨\cdot ,\cdot ⟩$ is an inner product if:

1. $⟨x,x⟩\ge 0$, $\forall x\in V$
2. $⟨x,x⟩=0⟺x=0$
3. $⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩$, $\forall x,y,z\in V$
4. $⟨\lambda x,y⟩=\lambda ⟨x,y⟩$, $\lambda \in \mathbb{K},x,y\in V$
5. $⟨x,y⟩=\overline{⟨y,x⟩}$ (the conjugate)

A vector space $V$ where an inner product is defined is called a vector space with inner product, and is denoted by $(V,⟨\cdot ,\cdot ⟩)$.

Example 1 (traditional ${\mathbb{R}}^n$)

In $({\mathbb{R}}^n,+,\cdot )$ we define the inner product of $u=(u_1,u_2,\dots ,u_n)$ and $v=(v_1,v_2,\dots ,v_n)$ as:

$$⟨u,v⟩=u_1{v}_1+u_2{v}_2+\cdots +u_n{v}_n$$

This operation satisfies the aforementioned properties of an inner product.

Example 2

Let $V=\mathcal{C}([a,b])=\{f:[a,b]\to \mathbb{R}\mid f\text{ is continuous in }[a,b]\}$

Then, we define:

$⟨f,g⟩:={\int }_a^{b}f(t)g(t)dt$!-

Inner product helps us define orthonormal and orthogonal sets.