How the concept of space is defined for Modular Form?

Question

The set of all functions of modular forms of weight $k$ is denoted by $M_k$. It is said in a document that $M_k$ is "clearly a vector space over $C$".

My question is if $M_k$ is a set of functions, how it becomes a vector space? Can anyone explain with an example?

Seems counter-intuitive!

Answer 1

The original concept of vectors applied specifically to vectors in a Euclidean space where each vector had a length and direction. This geometric concept was greatly generalized in the 19th century into the modern concept of a vector space. The Wikipedia article states

Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions

The $\,M_k\,$ is a good example of these function spaces. Several other natural examples appear in the Wikipedia article on this topic.