| Finite Dimensions | Infinite Dimensions | |---|---| | Vector $x \in \mathbbR^n$ | Function $f \in X$ (a space of functions) | | Matrix $A$ | Linear operator $T: X \to Y$ | | Solve $Ax = b$ | Solve $Tu = f$ | | Norm $|x|_2 = \sqrt\sum x_i^2$ | Norm $|f|_2 = \sqrt\int $ | | Convergence = componentwise | Convergence = uniform, pointwise, or in norm |
Bridging the gap from linear algebra to infinite-dimensional spaces without the fear factor a friendly approach to functional analysis pdf
The challenge: In infinite dimensions, not every Cauchy sequence converges unless you choose your space carefully. That's why we need and Hilbert spaces — they are the "complete" spaces where limits behave. | Finite Dimensions | Infinite Dimensions | |---|---|