Surjective function in the context of Injective function

In mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either the codomain or the image of a function.

A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution.

In linear algebra and functional analysis, a projection is a linear transformation ${\displaystyle P}$ from a vector space to itself (an endomorphism) such that ${\displaystyle P\circ P=P}$. That is, whenever ${\displaystyle P}$ is applied twice to any vector, it gives the same result as if it were applied once (i.e. ${\displaystyle P}$ is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.

In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set A may be indexed or labeled by means of the elements of a set J, then J is an index set. The indexing consists of a surjective function from J onto A, and the indexed collection is typically called an indexed family, often written as {A_j}_j∈J.

In mathematics, the range of a function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto. For any non-surjective function ${\displaystyle f:X\to Y,}$ the codomain ${\displaystyle Y}$ and the image ${\displaystyle {\tilde {Y}}}$ are different; however, a new function can be defined with the original function's image as its codomain, ${\displaystyle {\tilde {f}}:X\to {\tilde {Y}}}$ where ${\displaystyle {\tilde {f}}(x)=f(x).}$ This new function is surjective.

In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product.Non-trivial examples include rotations, reflections, and the Fourier operator.Unitary operators generalize unitary matrices.Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces.