TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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Bilinear forms on $V$ correspond linearly and bijectively to linear maps $V \to V^{*}$

Statement

The assignment

$B ⟼ B^{♭}, B^{♭} (v) (w) = B (v, w),$

is a linear bijection from the vector space of bilinear forms on $V$ to $L (V, V^{*})$ .

Facts & Assumptions

Given: An $F$ -vector space $V$ .

[L1]

A bilinear form is linear separately in both variables (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms).

[L2]

The algebraic dual $V^{*}$ consists of all linear maps $V \to F$ (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ).

[L3]

Linear maps form a vector space under pointwise addition and scalar multiplication (The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication).

Proof

technique · explicit inverse

1.1

For a bilinear $B$ , fixing $v$ makes $w \mapsto B (v, w)$ a member of $V^{*}$ by [L1] and [L2], and linearity in $v$ makes $B^{♭} : V \to V^{*}$ linear.

L1L2

1.2

Conversely, for $L \in L (V, V^{*})$ , define $B_{L} (v, w) = L (v) (w)$ . Linearity of $L$ gives linearity in $v$ , and $L (v) \in V^{*}$ gives linearity in $w$ , so $B_{L}$ is bilinear.

L1L2L3

2.1

Pointwise, $B_{B^{♭}} (v, w) = B (v, w)$ and $(B_{L})^{♭} (v) (w) = L (v) (w)$ , so the constructions are inverse. They also preserve addition and scalar multiplication pointwise.

step 1.1step 1.2L3algebra

3.1

Therefore $B \mapsto B^{♭}$ is a linear bijection. No finite-dimensionality assumption is used.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

H. Pinkham, Linear Algebra, Chapters 6–7 (standard reference, not scraped)

Bilinear forms on VV correspond linearly and bijectively to linear maps V→V∗V→V∗