Table 1 Seven essential pieces of information included in the map and its possible substructure

1

An inner product space is a vector space with an additional structure called the inner product function

Inner Product (IP) Space – Vector Space (VS)

2

An inner product function takes each ordered pair in a vector space V to a number in R

IP function – domain: V×V & codomain: R

3

An inner product function is a function which satisfy all following axioms: additivity, homogeneity, positivity, & symmetry

IP function – 4 axioms: additivity, homogeneity, positivity, & symmetry

4

Vector is an element of a vector space V

vector – IP Space (if the IP Space is connected to VS); or vector – VS

5

By using the inner product function of an inner product space, we can measure the orthogonal projection of a vector, the distance between two vectors, the length of a vector, and the angle between two vectors

IP Space – the measurements: orthogonal projection, distance between two vectors, length of a vector, angle between two vectors; or VS – the measurements

6

An inner product space is a vector space with an inner product function

IP Space – IP function; or VS – IP function (if the VS is connected to the IP Space)

7

Only a vector in R² and R³ that can be represented as a directed line segment, but not a vector in higher dimension

Vector – directed line segment; or the directed line segment node is not connected to any other nodes