Hilbert Space

linear-algebra

Definition

Hilbert Space

A Hilbert space $⟨\mathcal{H},⟨\cdot ,\cdot ⟩⟩$ is a real or complex inner product space whose induced metric space is complete.

Construction

A Hilbert space has three layers of structure.

1. It is a real or complex vector space $\mathcal{H}$.
2. It has an inner product $⟨\cdot ,\cdot ⟩$.
3. The inner product induces a norm

$$\Vert x\Vert =\sqrt{⟨x,x⟩}.$$

This norm induces the inner-product-induced metric space with metric

$$d(x,y)=\Vert x-y\Vert .$$

The space is Hilbert exactly when this inner-product-induced metric space is complete.