# COMPLEMENTARY TOPICS IN GEOMETRY

## Learning outcomes of the course unit

The course aims to provide the basic knowledge of the spectral theory of operators on real or complex inner product spaces, of the theory of bilinear and sesquilinear forms, of scalar and hermitian products, and of forms on real or complex inner product spaces.

The course aims to provide the basic knowledge of the spectral theory of operators on real or complex inner product spaces, of the theory of bilinear and sesquilinear forms, of scalar and hermitian products, and of forms on real or complex inner product spaces.

## Course contents summary

Dual space and the transpose of a linear transformation. Inner product spaces. Linear isometries and unitaty operators.

Linear functionals and adjoints. Spectral theory of operators on an inner product space: self-adjoint and normal operators. Bilinear and sesquilinear forms. Scalar and Hermitian products. Forms on inner product spaces.

Dual space and the transpose of a linear transformation. Inner product spaces. Linear isometries and unitaty operators.

Linear functionals and adjoints. Spectral theory of operators on an inner product space: self-adjoint and normal operators. Bilinear and sesquilinear forms. Scalar and Hermitian products. Forms on inner product spaces.

## Course contents

Dual of a vector space and the dual basis. The double dual of a vector space. Annihilator of a subset of a vector space. The transpose of a linear transformation. Real or complex inner product spaces: orthogonal complement of a linear subspace. Linear isometries and unitary operators: unitary and orthogonal matrices. Representation of a linear functional and adjoint of a linear operator. Self-adjoint operators: symmetric and hermitian operators. Spectral theorem for self-adjoint operators and for normal operators. Bilinear and sesquilinear forms: basis change, congruence, polar form. Scalar and hermitian products: existence of orthonormal bases. The Sylvester theorem: congruence criterion for symmetric matrices. Forms on inner product spaces: reduction to principal axes, simultaneous reduction of a pair of quadratic forms to a sum of squares.

Dual of a vector space and the dual basis. The double dual of a vector space. Annihilator of a subset of a vector space. The transpose of a linear transformation. Real or complex inner product spaces: orthogonal complement of a linear subspace. Linear isometries and unitary operators: unitary and orthogonal matrices. Representation of a linear functional and adjoint of a linear operator. Self-adjoint operators: symmetric and hermitian operators. Spectral theorem for self-adjoint operators and for normal operators. Bilinear and sesquilinear forms: basis change, congruence, polar form. Scalar and hermitian products: existence of orthonormal bases. The Sylvester theorem: congruence criterion for symmetric matrices. Forms on inner product spaces: reduction to principal axes, simultaneous reduction of a pair of quadratic forms to a sum of squares.

## Recommended readings

Marco Abate, GEOMETRIA, MacGraw-Hill, Milano, 1996.

Marco Abate, GEOMETRIA, MacGraw-Hill, Milano, 1996.

## Teaching methods

The theoretical topics of the course are presented during class lectures and illustrated with significant examples, applications and several exercises. Homework assignments are proposed during the course, which are then discussed in recitation sessions during class time.

The theoretical topics of the course are presented during class lectures and illustrated with significant examples, applications and several exercises. Homework assignments are proposed during the course, which are then discussed in recitation sessions during class time.

## Assessment methods and criteria

The final exam consists of a written part, where students are required to

solve some exercises, and of an oral part about the theoretical topics and

the applications discussed during the course.

The final exam consists of a written part, where students are required to

solve some exercises, and of an oral part about the theoretical topics and

the applications discussed during the course.