0
votes
0answers
44 views

Second derivative dot product

I was wondering whether I am correct that the second derivative of the dot product is this: Let $f:X \times X \rightarrow \mathbb{K}$ $(x_1,x_2) \mapsto \langle x_1,x_2 \rangle$ Then we have ...
2
votes
1answer
28 views

sufficient condition for being an integral factor

Let $ f: \mathbb {R}^m \rightarrow \mathbb {R}-\{0\} $ function $C^{\infty}$ class and $w$ a one-form $C^{\infty}$ class in $\mathbb {R}^m $. If $\alpha=w-\dfrac{1}{f}dx_{m+1} $ satisfies $\alpha ...
1
vote
1answer
100 views

Lipschitz condition in infinite dimensional vector spaces

If we have that $T:V \times W \rightarrow Y$ multilinear and $V,W$ are infinite-dimensional normed vector spaces.(the finite-dimensional proof is easy, since you can use compactness of the boundary ...
1
vote
1answer
292 views

Geometric meaning of Gram determinant

Let $v_1,v_2$ be vectors in $\mathbb{R}^4$. Let $M$ be the $2\times 4$ matrix with rows $v_1,v_2$ in this order. The Gram determinant of $M$ is defined as the determinant of the $2\times 2$ matrix ...
1
vote
1answer
248 views

A generalization of Lagrange identity

Let $k,n$ be positive integers, $k\le n$. Let $v_1,\cdots,v_k$ be vectors in $\mathbb{R}^n$. Let $M$ be the $k\times n$ matrix with rows $v_1,\cdots,v_k$ in this order. The Gram determinant of $M$ is ...
0
votes
1answer
75 views

A system of nonlinear equations

Does the following system of six simultaneous equations in eight variables $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4$ have solutions in $\mathbb{R}$? in $\mathbb{C}$? $$x_1y_2-x_2y_1=1$$ $$x_1y_3-x_3y_1=0$$ ...