📂Hilbert Space

Inner product spaces

Definition¹

Let’s consider $X$ as a vector space. For $\mathbf{x}, \mathbf{y}, \mathbf{z} \in X$ and $\alpha, \beta \in \mathbb{C}$(or $\mathbb{R}$), the following conditions satisfied by a function

$$ \langle \cdot , \cdot \rangle : X \times X \to \mathbb{C} $$

are defined as the inner product, and $\left( X, \langle \cdot ,\cdot \rangle \right)$ is called an inner product space.

• Linearity: $$\langle \alpha \mathbf{x} + \beta \mathbf{y} ,\mathbf{z} \rangle =\alpha \langle \mathbf{x},\mathbf{z}\rangle + \beta \langle \mathbf{y},\mathbf{z}\rangle$$
• Conjugate symmetry: $$\langle \mathbf{x},\mathbf{y} \rangle = \overline{ \langle \mathbf{y},\mathbf{x} \rangle}$$
• Positive-definiteness: $$\langle \mathbf{x},\mathbf{x} \rangle \ge 0 \quad \text{and} \quad \langle \mathbf{x},\mathbf{x} \rangle = 0\iff \mathbf{x}=0$$

Explanation

From linearity and conjugate symmetry, the following equation is obtained.

$$ \begin{align*} \langle \mathbf{x},\alpha \mathbf{y}+\beta \mathbf{z} \rangle =&\ \overline{\langle \alpha \mathbf{y}+\beta \mathbf{z} ,\mathbf{x} \rangle} \\ =&\ \overline{\alpha \langle \mathbf{y},\mathbf{x} \rangle +\beta \langle \mathbf{z},\mathbf{x} \rangle} \\ =&\ \overline{\alpha}\overline{\langle \mathbf{y},\mathbf{x} \rangle}+\overline{\beta} \overline{\langle \mathbf{z},\mathbf{x} \rangle} \\ =&\ \overline{\alpha}\langle \mathbf{x},\mathbf{y}\rangle + \overline{\beta} \langle \mathbf{x},\mathbf{z} \rangle \end{align*} $$

This means it is antilinear regarding the second element. In physics, engineering, etc., the inner product might be defined slightly differently. For example, it can be defined as being antilinear with respect to the first component, and linear with the second. Meanwhile, in an inner product space, the Cauchy-Schwarz inequality is established as below.

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Let’s say $(X, \langle \cdot ,\cdot \rangle)$ is an inner product space. Then, the following inequality holds, which is called the Cauchy-Schwarz inequality.

$$ \left| \langle \mathbf{x},\mathbf{y} \rangle \right| \le \langle \mathbf{x},\mathbf{x} \rangle^{1/2} \langle \mathbf{y},\mathbf{y} \rangle ^{1/2},\quad \forall \mathbf{x},\mathbf{y} \in X $$

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Moreover, an norm can be defined from an inner product.

$$ \left\| \mathbf{x} \right\| := \sqrt{\langle \mathbf{x},\mathbf{x} \rangle},\quad \mathbf{x}\in X $$

This naturally derived norm from an inner product is also referred to as the associated norm. Also, given a norm, the distance can be defined from the norm, thus enabling discussion on the property of the metric space, which is completeness. A complete inner product space is called a Hilbert space.

Properties

Cauchy-Schwarz inequality: For any $\mathbf{x},\mathbf{y}\in X$,

$$ \left| \langle \mathbf{x},\mathbf{y} \rangle \right| \le \left\| \mathbf{x} \right\| \left\| \mathbf{y} \right\| $$

Parallelogram law: For any $\mathbf{x},\mathbf{y}\in X$,

$$ \left\| \mathbf{x} + \mathbf{y} \right\|^{2} + \left\| \mathbf{x} - \mathbf{y} \right\|^{2} = 2 \left( \left\| \mathbf{x} \right\| ^{2}+ \left\| \mathbf{y} \right\| ^{2} \right) $$

The polarization identity in a complex vector space: For a complex inner product space $X$ and any $\mathbf{x},\mathbf{y}\in X$,

$$ \langle \mathbf{x},\mathbf{y} \rangle = \frac{1}{4} \Big( \left\| \mathbf{x} + \mathbf{y} \right\|^{2} - \left\| \mathbf{x} - \mathbf{y} \right\|^{2} + i \left( \left\| \mathbf{x} + iy \right\|^{2} - \left\| \mathbf{x} - iy \right\|^{2} \right) \Big) $$

The polarization identity in a real vector space: For a real inner product space $X$ and any $\mathbf{x},\mathbf{y}\in X$,

$$ \langle \mathbf{x},\mathbf{y}\rangle = \frac{1}{4} \left( \left\| \mathbf{x}+\mathbf{y} \right\|^{2} - \left\| \mathbf{x}-\mathbf{y} \right\| ^{2} \right) $$

Norm versus inner product: For any $\mathbf{x} \in X$,

$$ \left\| \mathbf{x} \right\| =\sup \left\{ \left| \langle \mathbf{x},\mathbf{y} \rangle \right| : \mathbf{y}\in X, \left\| \mathbf{y} \right\| =1 \right\} $$