Functional Analysis
Functional Analysis in GATE MA covers normed and Banach spaces, Hilbert spaces, bounded operators, and the four big theorems. Expect norm computations, dual-space arguments, and compactness questions.
Core rules
- A normed space is Banach if complete; Hilbert if the norm comes from an inner product and it is complete.
- Bounded linear T is equivalent to continuous: ‖T‖=sup_{‖x‖≤1}‖Tx‖<∞.
- Hahn–Banach extends bounded linear functionals; open mapping, closed graph, and uniform boundedness require Banach spaces.
- Riesz representation: every bounded linear functional on a Hilbert space H is ⟨·,y⟩ for a unique y∈H.
- Compact operators map bounded sets to relatively compact sets; they have discrete spectrum accumulating only at 0.
- The dual space X* consists of bounded linear functionals with ‖f‖=sup_{‖x‖≤1}|f(x)|.
Worked example
On C[0,1] with the sup norm, define Tf=∫0^1 f(x)dx. Then |Tf|≤∫0^1|f|≤‖f‖∞, so ‖T‖≤1. Taking f≡1 gives Tf=1=‖f‖∞, hence ‖T‖=1. T is bounded, so continuous.
Traps
- Do not use open mapping or closed graph without completeness; they fail in incomplete normed spaces.
- The closed graph theorem says a closed graph implies boundedness for Banach spaces, not for arbitrary normed spaces.
- In a Hilbert space, the Riesz map is conjugate-linear in the complex case; ⟨x,αy⟩=ᾱ⟨x,y⟩.
- A compact operator need not have finite rank; the identity on an infinite-dimensional space is not compact.