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PAPER X: FUNCTIONAL ANALYSIS

Unit I
: Banach spaces – The definition and some examples – Continuous linear transformations – The Hahn-Banach theorem – The natural imbedding of N in N** - The open mapping problem- The conjugate of an operator.

Unit II
: Hilbert spaces – The definition and some simple properties – Orthogonal complements - Orthonormal sets- The Conjugate space H* - The adjoint of an operator – Self-adjoint operators – Normal and unitary operators.

Unit III
: Projections -Matrices – Determinants and the spectrum of an operator – The spectral theorem.

Unit IV
: The definition and some examples of Banach algebra – Regular and singular elements – Topological divisors of zero – The spectrum – The formula for the spectral radius-The radical and semisimplicity

Unit V
: The Gelfand mapping-Applications of the formula r(x)=lim||xn|| (1/n) – Involutions in Banach algebras- The Gelfand-Neumark theorem- Ideals in C(X) and the Banach-Stone theorem.

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