This post tests various functionality.

Mathematics

Let’s start with some inline math eiπ+1=0e^{i\pi} + 1 = 0.

Here’s a display equation—the Cauchy-Schwarz inequality in a :

∣⟨x,y⟩∣2≤⟨x,x⟩⋅⟨y,y⟩|\langle x, y \rangle|^2 \leq \langle x, x \rangle \cdot \langle y, y \rangle

The definition of the operator norm in a :

∥T∥=sup⁡∥x∥≤1∥Tx∥=sup⁡x≠0∥Tx∥∥x∥\|T\| = \sup_{\|x\| \leq 1} \|Tx\| = \sup_{x \neq 0} \frac{\|Tx\|}{\|x\|}

Code Blocks

Python

import numpy as np
from scipy.linalg import norm

def gram_schmidt(vectors):
    """Orthonormalize a set of vectors using Gram-Schmidt."""
    basis = []
    for v in vectors:
        w = v - sum(np.dot(v, b) * b for b in basis)
        if norm(w) > 1e-10:
            basis.append(w / norm(w))
    return np.array(basis)

# Example: orthonormalize R^3 vectors
V = np.array([[1, 1, 0], [1, 0, 1], [0, 1, 1]], dtype=float)
Q = gram_schmidt(V)
print(Q @ Q.T)  # Should be approximately identity

/-- A vector space over a field K -/
class VectorSpace (K : Type*) (V : Type*) [Field K] extends AddCommGroup V, Module K V

/-- The dual space of a vector space -/
def DualSpace (K : Type*) (V : Type*) [Field K] [AddCommGroup V] [Module K V] :=
  V →ₗ[K] K

theorem dual_dual_iso (K : Type*) (V : Type*) [Field K] [AddCommGroup V] [Module K V]
    [FiniteDimensional K V] :
    FiniteDimensional.finrank K (DualSpace K (DualSpace K V)) = FiniteDimensional.finrank K V := by
  simp [DualSpace, LinearMap.finrank_linearMap]

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