definition of tensors space w.r.t. field and k vectors spaces and vectors space over field
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of linear map.
Target Context
- The reader will have a definition of tensors space with respect to field and k vectors spaces and vectors space over field.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V_1, ..., V_k, W\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\(*L (V_1, ..., V_k: W)\): \(= \{f: V_1, ..., V_k \to W: \forall j \in \{1, ..., k\}, \forall v_1 \in V_1, ..., \forall v_k \in V_k, \forall v'_j \in V_j, \forall r, r' \in F (f (v_1, ..., r v_j + r' v'_j, ..., v_k) = r f (v_1, ..., v_j, ..., v_k) + r' f (v_1, ..., v'_j, ..., v_k))\}\), \(\in \{\text{ the } F \text{ vectors spaces }\}\) with the addition and the scalar multiplication specified below
//
Conditions:
\(\forall f_1, f_2 \in L (V_1, ..., V_k: W) ((f_1 + f_2) (v_1, ..., v_k) = f_1 (v_1, ..., v_k) + f_2 (v_1, ..., v_k))\)
\(\land\)
\(\forall f \in L (V_1, ..., V_k: W), \forall r \in F ((r f) (v_1, ..., v_k) = r (f (v_1, ..., v_k))\)
//
2: Note
In other words, \(f\) is a multi-linear map.
\(F\) needs to be the same for all the vectors spaces, because otherwise the linearity would not make sense: "\(r f (v_1, ..., v_j, ..., v_k)\)" would not make sense if \(W\) was not an \(F\) vectors space, so, each \(V_j\) needs to be over the field of \(W\). In fact, there can be the argument that the field of \(W\) does not need to be exactly the filed of \(V_j\) but needs to be an extension of the field of \(V_j\), which may be indeed possible, but we do not see any immediate necessity to allow that case complicating the situation.
Let us see that \(L (V_1, ..., V_k: W)\) is indeed an \(F\) vectors space.
1) for any elements, \(f_1, f_2 \in L (V_1, ..., V_k: W)\), \(f_1 + f_2 \in L (V_1, ..., V_k: W)\) (closed-ness under addition): \(f_1 + f_2\) is \(: V_1, ..., V_k \to W\); \((f_1 + f_2) (v_1, ..., r v_j + r' v'_j, ..., v_k) = f_1 (v_1, ..., r v_j + r' v'_j, ..., v_k) + f_2 (v_1, ..., r v_j + r' v'_j, ..., v_k) = r f_1 (v_1, ..., v_j, ..., v_k) + r' f_1 (v_1, ..., v'_j, ..., v_k) + r f_2 (v_1, ..., v_j, ..., v_k) + r' f_2 (v_1, ..., v'_j, ..., v_k) = r f_1 (v_1, ..., v_j, ..., v_k) + r f_2 (v_1, ..., v_j, ..., v_k) + r' f_1 (v_1, ..., v'_j, ..., v_k) + r' f_2 (v_1, ..., v'_j, ..., v_k) = r (f_1 (v_1, ..., v_j, ..., v_k) + f_2 (v_1, ..., v_j, ..., v_k)) + r' (f_1 (v_1, ..., v'_j, ..., v_k) + f_2 (v_1, ..., v'_j, ..., v_k)) = r (f_1 + f_2) (v_1, ..., v_j, ..., v_k) + r' (f_1 + f_2) (v_1, ..., v'_j, ..., v_k)\).
2) for any elements, \(f_1, f_2 \in L (V_1, ..., V_k: W)\), \(f_1 + f_2 = f_2 + f_1\) (commutativity of addition): \((f_1 + f_2) (v_1, ..., v_k) = f_1 (v_1, ..., v_k) + f_2 (v_1, ..., v_k) = f_2 (v_1, ..., v_k) + f_1 (v_1, ..., v_k) = (f_2 + f_1) (v_1, ..., v_k)\).
3) for any elements, \(f_1, f_2, f_3 \in L (V_1, ..., V_k: W)\), \((f_1 + f_2) + f_3 = f_1 + (f_2 + f_3)\) (associativity of additions): \(((f_1 + f_2) + f_3) (v_1, ..., v_k) = (f_1 + f_2) (v_1, ..., v_k) + f_3 (v_1, ..., v_k) = f_1 (v_1, ..., v_k) + f_2 (v_1, ..., v_k) + f_3 (v_1, ..., v_k) = f_1 (v_1, ..., v_k) + (f_2 (v_1, ..., v_k) + f_3 (v_1, ..., v_k)) = f_1 (v_1, ..., v_k) + (f_2 + f_3) (v_1, ..., v_k) = (f_1 + (f_2 + f_3)) (v_1, ..., v_k)\).
4) there is a 0 element, \(0 \in L (V_1, ..., V_k: W)\), such that for any \(f \in L (V_1, ..., V_k: W)\), \(f + 0 = f\) (existence of 0 vector): the \(0\) map, \(f_0 \in L (V_1, ..., V_k: W)\), is \(0\), because \((f + f_0) (v_1, ..., v_k) = f (v_1, ..., v_k) + f_0 (v_1, ..., v_k) = f (v_1, ..., v_k) + 0 = f (v_1, ..., v_k)\).
5) for any element, \(f \in L (V_1, ..., V_k: W)\), there is an inverse element, \(f' \in L (V_1, ..., V_k: W)\), such that \(f' + f = 0\) (existence of inverse vector): \(- f \in L (V_1, ..., V_k: W)\) is \(f'\), because \((- f + f) (v_1, ..., v_k) = - f (v_1, ..., v_k) + f (v_1, ..., v_k) = 0 = f_0 (v_1, ..., v_k)\).
6) for any element, \(f \in L (V_1, ..., V_k: W)\), and any scalar, \(r \in F\), \(r . f \in L (V_1, ..., V_k: W)\) (closed-ness under scalar multiplication): \(r . f\) is \(: V_1, ..., V_k \to W\); \((r . f) (v_1, ..., s v_j + s' v'_j, ..., v_k) = r f (v_1, ..., s v_j + s' v'_j, ..., v_k) = r (s f (v_1, ..., v_j, ..., v_k) + s' f (v_1, ..., v'_j, ..., v_k)) = s r f (v_1, ..., v_j, ..., v_k) + s' r f (v_1, ..., v'_j, ..., v_k) = s (r f) (v_1, ..., v_j, ..., v_k) + s' (r f) (v_1, ..., v'_j, ..., v_k)\).
7) for any element, \(f \in L (V_1, ..., V_k: W)\), and any scalars, \(r_1, r_2 \in F\), \((r_1 + r_2) . f = r_1 . f + r_2 . f\) (scalar multiplication distributability for scalars addition): \(((r_1 + r_2) . f) (v_1, ... v_k) = (r_1 + r_2) f (v_1, ... v_k) = r_1 f (v_1, ... v_k) + r_2 f (v_1, ... v_k) = (r_1 f) (v_1, ... v_k) + (r_2 f) (v_1, ... v_k) = (r_1 . f + r_2 . f) (v_1, ... v_k)\).
8) for any elements, \(f_1, f_2 \in L (V_1, ..., V_k: W)\), and any scalar, \(r \in F\), \(r . (f_1 + f_2) = r . f_1 + r . f_2\) (scalar multiplication distributability for vectors addition): \((r . (f_1 + f_2)) (v_1, ... v_k) = r (f_1 + f_2) (v_1, ... v_k) = r (f_1 (v_1, ... v_k) + f_2 (v_1, ... v_k)) = r f_1 (v_1, ... v_k) + r f_2 (v_1, ... v_k) = (r f_1) (v_1, ... v_k) + (r f_2) (v_1, ... v_k) = (r . f_1 + r . f_2) (v_1, ... v_k)\).
9) for any element, \(f \in L (V_1, ..., V_k: W)\), and any scalars, \(r_1, r_2 \in F\), \((r_1 r_2) . f = r_1 . (r_2 . f)\) (associativity of scalar multiplications): \(((r_1 r_2) . f) (v_1, ... v_k) = (r_1 r_2) f (v_1, ... v_k) = r_1 (r_2 f (v_1, ... v_k)) = r_1 ((r_2 f) (v_1, ... v_k)) = (r_1 . (r_2 . f)) (v_1, ... v_k)\).
10) for any element, \(f \in L (V_1, ..., V_k: W)\), \(1 . f = f\) (identity of 1 multiplication): \((1 . f) (v_1, ... v_k) = 1 f (v_1, ... v_k) = f (v_1, ... v_k)\).
A typical example is \(L (V_1, ..., V_k: F)\): \(F\) is an \(F\) vectors space.
A more typical example is \(L (V: F) := V^*\), called "the covectors space of \(V\)" or "the dual space of \(V\)".
Another typical example is \(T^p_q (V) := L (V^*, ..., V^*, V, ..., V: F)\), where there are \(p\) \(V^*\) s and \(q\) \(V\) s.
\(V^* = T^0_1 (V)\).
Each element of \(L (V_1, ..., V_k: W)\) is called "tensor".
Each element of \(T^0_l (V) := L (V, ..., V: F)\) is called "multicovector".
Each element of \(V^*\) is called "covector".
As \(L (V_1, ..., V_k: W)\) is an \(F\) vectors space, it admits a basis, and any tensor has the components with respect to the basis. Such a set of components is sometimes called "tensor", but the set of components is exactly the set of the components of the tensor with respect to the basis, not "tensor".
