Functional Analysis – Android

ivan gabrovski

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About this app

Educational application with 7 interactive modules:

1. **Norms in ℝⁿ** – ℓᵖ norms, Hölder, Minkowski, Cauchy–Schwarz, unit spheres for p = 1, 2, ∞ and arbitrary p.
2. **Functional spaces** – ‖f‖₁, ‖f‖₂, ‖f‖_p, ‖f‖∞ on [a,b], embeddings Lᵖ ⊂ L^q, ‖f‖_p → ‖f‖∞.
3. **Gram–Schmidt in L²_w** – orthonormal polynomials with weight (Legendaire, Chebyshev, Lager), projection, Bessel, Pythagoras.
4. **Fourier series** – coefficients, Bessel/Parseval, L² and uniform error, Gibbs phenomenon, spectrum.
5. **Linear operators** – ‖A‖₁, ‖A‖∞, ‖A‖₂, Frobenius, singular numbers, κ₂, Gelfand's formula for ρ(A), Neumann series, image of the unit circle.
6. **Shrinking images** – Banach theorem, q = max|g′|, iterations, prior/a posteriori estimation, spider web diagram.
7. **Fredholm integral equations of the second kind** – contraction constants (C and L²), successive approximations, Nyström's method, Fredholm alternative.

## Languages
English (basic), Bulgarian, German, French and Spanish.
- Interface and theory: `values/` (English), `values-bg`, `values-de`, `values-fr`, `values-es`.
- Calculation reports (pure Java) are translated from `core/Tr.java`; each Engine registers
its strings in a static block, and `ModuleActivity` synchronizes the language with the resources.
- The language is changed from the 🌐 icon on the home screen via `AppCompatDelegate.setApplicationLocales`
and does not depend on the device language. When first launched, the application is in English.
The selection is also preserved on Android 12 and older (`AppLocalesMetadataHolderService`);
On Android 13+, the language can also be changed from the system settings ("App languages").

## Expression syntax
`+ - * / ^`, implicit multiplication (`2x`, `3sin(x)`, `x t`), variables `x`, `t`,
констани `pi`, `e`, `inf`, бизнеси `sin cos tan asin acos atan sinh cosh tanh exp ln log sqrt abs sgn floor ceil`.
Number fields also accept expressions: `2pi`, `sqrt(2)`, `1e-8`.

Calculations are performed in a background thread; results are automatically recalculated after screen rotation.