A continuous function is a mathematical function that has no sudden or discontinuous changes. This means that the graph of the function can be drawn without lifting the pen from the paper.
Pointwise convergence means that for each point in the domain, the sequence of function values converges to the corresponding value of the limit function. Uniform convergence means that for any small interval, the sequence of function values eventually gets closer and closer to the values of the limit function.
A vector space is a set of objects called vectors, which can be added together and multiplied by numbers, called scalars. Its main properties include closure under addition and scalar multiplication, associativity, commutativity, distributivity, and the existence of an identity element and inverse elements.
To prove that a set is a vector space, you must show that it satisfies all of the main properties of a vector space, such as closure under addition and scalar multiplication. You can do this by demonstrating that the operations of addition and scalar multiplication are well-defined, and that the set contains an identity element and inverse elements.
The dimension of a vector space is the number of vectors in a basis for that space. It is determined by finding the minimum number of linearly independent vectors that span the entire vector space. This can be done by reducing the matrix of the vectors to its reduced row echelon form and counting the number of nonzero rows.