Inner Product and Space

• categories: Functional analysis, Definition

Inner Product Space An inner product space (or pre-Hilbert space) is a vector space $V$ over the field $\mathbb{R}$ or $\mathbb{C}$ equipped with an inner product. The inner product provides a way to define geometric concepts such as angles, lengths, and orthogonality within the space.

Definition

An inner product on a vector space $V$ is a function $⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{F}$ (where $\mathbb{F}$ is $\mathbb{R}$ or $\mathbb{C}$) that satisfies the following properties for all $u,v,w\in V$ and $\alpha \in \mathbb{F}$:

1. Linearity (in the first argument):

   $$⟨\alpha u+v,w⟩=\alpha ⟨u,w⟩+⟨v,w⟩.$$

   If $\mathbb{F}=\mathbb{C}$, the inner product is conjugate-linear in the first argument and linear in the second.

2. Conjugate Symmetry:

   $$⟨u,v⟩=\overline{⟨v,u⟩}.$$

   If $\mathbb{F}=\mathbb{R}$, this reduces to $⟨u,v⟩=⟨v,u⟩$.

3. Positive Definiteness:

   $$⟨v,v⟩\ge 0,\text{and}⟨v,v⟩=0⟺v=0.$$

Examples

1. Euclidean Space: In ${\mathbb{R}}^n$, the standard inner product is given by: $$⟨x,y⟩=\sum _{i=1}^n{x}_i{y}_i.$$
2. Complex Inner Product Space: In ${\mathbb{C}}^n$, the standard inner product is: $$⟨x,y⟩=\sum _{i=1}^n{x}_i\overline{y_i}.$$
3. Function Spaces: In $L^2([a,b])$, the inner product of functions $f$ and $g$ is: $$⟨f,g⟩={\int }_a^{b}f(x)\overline{g(x)}dx.$$

Properties

1. Norm Induced by Inner Product: The inner product induces a Norm $\Vert v\Vert =\sqrt{⟨v,v⟩}$, which satisfies the properties of a vector norm.
2. Cauchy–Schwarz Inequality: For any $u,v\in V$, $$|⟨u,v⟩|\le \Vert u\Vert \cdot \Vert v\Vert .$$
3. Triangle Inequality: For any $u,v\in V$, $$\Vert u+v\Vert \le \Vert u\Vert +\Vert v\Vert .$$
4. Orthogonality: Vectors $u$ and $v$ are orthogonal if $⟨u,v⟩=0$.