A function space is a set of functions that satisfy certain conditions. Polynomials with real coefficients are not functions, so they cannot form a function space.

A normed space is a vector space that is equipped with a norm, which is a function that assigns a non-negative real number to each vector in the space. The norm satisfies certain properties, including completeness, closure under addition and scalar multiplication, and the triangle inequality.

A Banach space is a complete normed space. The space of continuous functions on the interval [0, 1] is a complete normed space, so it is a Banach space.

A Hilbert space is a complete inner product space. The space of square-integrable functions on the interval [0, 1] is a complete inner product space, so it is a Hilbert space.

A compact operator is a bounded linear operator whose spectrum is a compact set. It also has a finite-dimensional range.

A Fredholm operator is a bounded linear operator whose index is zero. Its spectrum is also a closed set.

A trace class operator is a compact operator whose trace is finite. Its spectrum is also a discrete set.

A Hilbert-Schmidt operator is a compact operator whose Hilbert-Schmidt norm is finite. Its spectrum is also a compact set.

A positive operator is a self-adjoint operator whose spectrum is a subset of the non-negative real numbers. It also has a positive trace.

A projection operator is a self-adjoint operator whose range is a closed subspace. Its kernel is also a closed subspace.

A unitary operator is a bounded linear operator whose inverse is also unitary. Its spectrum is also a subset of the unit circle.

A normal operator is a bounded linear operator that commutes with its adjoint. Its spectrum is also a closed set.

A self-adjoint operator is a normal operator whose spectrum is a subset of the real numbers. Its eigenvectors are also orthogonal.

A bounded linear operator is a continuous linear operator whose range is a closed subspace. Its kernel is also a closed subspace.

A closed linear operator is a linear operator whose graph is a closed subspace. Its range is also a closed subspace, and its kernel is a closed subspace.