Functional Analysis

Mathematical Principles & Theorems

Investigates infinite-dimensional vector spaces. Banach spaces are complete normed vector spaces \((X, \|\cdot\|)\); Hilbert spaces \(\mathcal{H}\) are complete inner product spaces with norm \(\|f\| = \sqrt{\langle f, f \rangle}\). The \(L^2[a, b]\) inner product is \(\langle f, g \rangle = \int_a^b f(x)\overline{g(x)}dx\). Riesz representation theorem establishes an isometric isomorphism between \(\mathcal{H}\) and its continuous dual space \(\mathcal{H}^*\). Orthogonal expansions decompose functions over orthonormal bases \(f = \sum \langle f, e_n \rangle e_n\).

Operating Instructions

• Select an orthogonal function basis: Fourier Sinusoids, Legendre Polynomials, or Chebyshev Polynomials.
• Input test functions \(f(x)\) and \(g(x)\) in the function boxes.
• Click Compute to evaluate inner product \(\langle f, g \rangle\), \(L^2\) norms, and orthogonality angle.
• Inspect the basis projection synthesis and energy spectrum readouts.