### The Theory of Quadratic Forms in Infinitely Many Variables and Applications

American Mathematical Society Colloquium Lecture Series, Madison, Wisconsin, September 6-10, 1927

This outline of Wheeler's lectures was published in the Bulletin of the American Mathematical Society, Vol. 33, (1927), 664-665.

*Spaces of infinitely many dimensions.*

Vectors. Fundamental quadratic form. Scalar product. Orthogonality. Construction of coordinate systems corresponding to a given fundamental form. Properties of coordinate systems in euclidean spaces. Properties of coordinate systems in non-euclidean spaces. Connection between spaces of different types and classes of functions. General coordinate systems.

*Infinite matrices and linear transformations.*

Matrices, bilinear forms, and linear transformations. Limitedness with respect to Hilbert space. Limitedness with respect to other spaces. Complete continuity in the Hilbert sense. Extension. Infinite determinants. Some sufficient conditions for the alternative theorems. The Fredholm determinant. Principal solutions of homogeneous equations.

*Reduction of quadratic and bilinear forms.*

Nature of invariants of linear transformations in the symmetric case. A necessary condition for the existence of invariants in the symmetric case. Carleman's sufficient condition and extension in the symmetric case. Symmetrizable cases. Semi-symmetrizable cases. Simultaneous reduction of two quadratic forms. Solutions of more general linear equations.

*Application to the theory of linear functional equations*.

Integral equations with continuous symmetric kernels. integral equations with singular symmetric kernels. Symmetrizable kernels. Semi-symmetrizable kernels. integro-differential equations. Systems of integral equations. Linear functional equations of various types.

*Application to the theory of linear differential equations*.

Ordinary linear differential equations, orthogonal and polar. Singular linear differential equations. Elliptic differential equations. Hyperbolic differential equations. The Bliss system of differential equations.