Automatic Theorem Proofs

Geometry of interaction

Functional Analysis Theorems:

1. Riesz Representation Theorem: $\forall f \in \mathcal{H}', \exists y \in \mathcal{H} \text{ such that } f(x) = \langle x, y \rangle \text{ for all } x \in \mathcal{H}$

2. Open Mapping Theorem (Banach-Steinhaus Theorem): $\text{If } T:X \to Y \text{ is a bounded linear operator between Banach spaces, then } T(X) \text{ is open in } Y$

3. Closed Graph Theorem: $\text{A linear operator } T:X \to Y \text{ between Banach spaces is closed if and only if its graph } {(x, Tx) \mid x \in X} \text{ is closed in } X \times Y.$

4. Hahn-Banach Theorem: $\text{For any subspace } Y \subset X \text{ and any linear functional } f_0 : Y \to \mathbb{K}, \text{ there exists an extension } f : X \to \mathbb{K} \text{ such that } f|_Y = f_0 \text{ and } |f| = |f_0|$

5. Spectral Theorem: $\text{Let } A \text{ be a self-adjoint (or normal) operator on a Hilbert space } \mathcal{H}.$ $\text{ Then, there exists a spectral measure } E \text{ such that}$ $A = \int_{\sigma(A)} \lambda , dE(\lambda),$ $\text{where } \sigma(A) \text{ is the spectrum of } A.$

Linear Algebra

1. Gradient Descent

   Gradient descent is a simple and effective algorithm for finding the minimum of a function. It is often used in machine learning to train models.

   $\theta_n+1 = \theta_n - \eta \nabla J(\theta_n)$ where:

   $\theta$ is the parameter vector , $J(\theta)$ is the loss function , $\eta$ is the learning rate, $\nabla J(\theta)$ is the gradient of the loss function with respect to the parameters 2.