riesz representation theorem

Topics

• functional analysis
• vector inner product

Core idea: In any vector inner product space, a linear functional (linear maps to scalars) can be represented as an inner product with some fixed vector. Formally, for a vector space $V$ with inner product $⟨\cdot ,\cdot ⟩$, any linear functional $\varphi :V\to F$ can be written as:

$$\varphi (\mathbf{x})=⟨\mathbf{z},\mathbf{x}⟩$$

for some unique $\mathbf{z}\in V$

This theorem connects algebraic (functionals) and geometric (inner products) concepts

Proof sketch:

1. Take orthonormal basis $\{{\mathbf{u}}_k\}$ of $V$
2. Expand any vector $\mathbf{x}=\sum x_k{\mathbf{u}}_k$
3. By linearity: $$\varphi (\mathbf{x})=\sum x_k\varphi ({\mathbf{u}}_k)$$
4. This matches $⟨\mathbf{z},\mathbf{x}⟩$ if we choose: $$\mathbf{z}=\sum \overline{\varphi ({\mathbf{u}}_k)}{\mathbf{u}}_k$$

Finite vs infinite dimensions

• In finite dimensions, always holds
• In infinite dimensions (hilbert spaces), requires completeness

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