KindiakMath

Functional Analysis

Here are my write-ups on the theory of functional analysis and some non-trivial exercises therein. Functional analysis is linear algebra in infinitely many dimensions: functions become vectors, integrals become inner products, and differential and integral equations become linear maps. Measure theory supplies the spaces, and the theory repays the favour, as conditional expectation turns out to be an orthogonal projection. The goal of this set of posts is to build up to the four big theorems on Banach spaces and the spectral theorem for compact operators, and then to step beyond norms altogether: topological vector spaces make room for distributions, generalised functions that can be differentiated as often as we like. Enjoy!

Theory

Banach Spaces

  1. The Lebesgue Spaces
  2. Setting the Norm
  3. Filling the Gaps
  4. Within Bounds
  5. The Baire Necessities [TBC]

Hilbert Spaces and Operators

  1. Right Angles [TBC]
  2. Representation Matters [TBC]
  3. Good Vibrations [TBC]
  4. Mirror, Mirror [TBC]
  5. Seeing the Spectrum [TBC]

Duality and Topological Vector Spaces

  1. The Other Side [TBC]
  2. Double Vision [TBC]
  3. Beyond Norms [TBC]
  4. Separation Anxiety [TBC]
  5. Compact at Last [TBC]

Distributions

  1. Testing, Testing [TBC]
  2. Smoothing Things Over [TBC]
  3. Tempered Transforms [TBC]
  4. Weak Solutions [TBC]

Exercises

Banach Spaces

  1. Mind the Gap
  2. Vanishing Acts
  3. Converging Opinions
  4. Shifting Gears [TBC]
  5. Corners Everywhere [TBC]
  6. Breaking Waves [TBC]
  7. Measuring Up [TBC]

Hilbert Spaces and Operators

  1. Line of Best Fit [TBC]
  2. Best Estimates [TBC]
  3. Straightening Out [TBC]
  4. Basel Revisited [TBC]
  5. Step by Step [TBC]
  6. Running Totals [TBC]
  7. Kernels of Truth [TBC]
  8. Strings Attached [TBC]

Duality and Topological Vector Spaces

  1. Hidden Partners [TBC]
  2. Pushing the Limit [TBC]
  3. No Reflection [TBC]
  4. One Norm Is Not Enough [TBC]
  5. Nothing on the Other Side [TBC]
  6. Points of Certainty [TBC]

Distributions

  1. Infinitely Sharp [TBC]
  2. Cancelling Infinities [TBC]
  3. Feeling the Heat [TBC]
  4. Point Charges [TBC]
  5. All Roads Lead to Gauss [TBC]
  6. Point Charges [TBC]
  7. Hidden Continuity [TBC]