Show that the \(k\)-forms on \(M\) form a vector space (infinite-dimensional if \(k\) does not exceed the dimension of \(M\) ).

A differential form is an important concept in the study of calculus on manifolds, a space that generalizes the notion of curves and surfaces. Differential forms have applications ranging from physics to geometry and topology. They provide a way of extending the concept of integration to various types of manifolds. A differential form can be understood as a generalized function that, instead of associating a single number to each point on a manifold, associates a value to each small piece of the manifold.

Differential forms are particularly useful for expressing the properties of curves, surfaces, and higher-dimensional manifolds in a way that is independent of coordinates. Therefore, they are a key part of the modern language of calculus and play a central role in the field of differential geometry.

Simply put, a manifold is a space that locally resembles Euclidean space, meaning that each point has a neighborhood that is similar to an open space in Euclidean space. Manifolds serve as a playground for physics and geometry, where concepts like distance, angles, and curvature can be explored. They can have any number of dimensions and come in a variety of shapes and sizes. A circle, the surface of a sphere, and the universe itself can all be thought of as manifolds.

Mathematically, manifolds are studied using calculus and algebraic topology, providing a means to analyze and understand the complex structures and shapes that can exist in multi-dimensional spaces. In essence, a manifold is the setting for the differential forms we use to capture the essence of these spaces.

The vector space axioms form the backbone of linear algebra and are essential for understanding the behavior of different mathematical structures. To be considered a vector space, a set along with two operations (addition and scalar multiplication) must satisfy eight specific requirements. These include closure under addition and scalar multiplication, associativity, and commutativity of addition, existence of an identity and inverse element for addition, and distributivity of scalar multiplication over vector addition and field addition. These properties ensure that vector spaces have a well-structured and predictable behavior. When dealing with abstract entities like differential forms, the vector space axioms allow us to manipulate these objects mathematically, ensuring that our equations and formulas make sense within the given framework.

The wedge product is a mathematical operation used in the context of differential forms, playing a crucial role in defining the exterior algebra of forms on a manifold. It's analogous to the cross product in three dimensions, but it extends to arbitrary dimensions. This operation is anticommutative, meaning that swapping the order of two 1-forms changes the sign of the product.

The wedge product allows the combination of differential forms to generate new forms of higher degree, enabling a powerful and flexible system for calculus on manifolds. It is particularly important for defining integrals over surfaces or volumes in multidimensional space and for formulating equations in physics such as those found in electromagnetism and general relativity.

Tangent vectors are geometric objects that represent the direction and rate of speed of a curve at a point. Just like a vector on a plane points in a certain direction, a tangent vector on a manifold illustrates how one might 'move' along the manifold if traveling in the direction of the curve at a given point. They form the basis for the tangent space at a point on a manifold, which is, in essence, a local approximation of the manifold around that point.

When applied to differential forms, tangent vectors are the entities upon which forms act. In a k-form, we input k such tangent vectors, and the form gives us a real number, often representing some type of volume or other geometric property. This interaction provides a bridge between abstract algebra and tangible geometric intuition.