Intereting Posts

How likely is it for a randomly picked number to be larger than all previously chosen numbers?
How many closed subsets of $\mathbb R$ are there up to homeomorphism?
Trigonometric Equation $\sin x=\tan\frac{\pi}{15}\tan\frac{4\pi}{15}\tan\frac{3\pi}{10}\tan\frac{6\pi}{15}$
Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$
Factor a polynomial
Textbook for Projective Geometry
Prove that if $a\equiv b \pmod m $ , then $a \bmod m = b \bmod m$
“A Function Can't Be Odd&Even” They said, Right?
$i'=i^{-1} \bmod p$, prove or disprove that $\lim_{p\to \infty}\dfrac{1}{p^3}\sum_{i=1}^{p-1}ii'=\frac{1}4$
Convergence and closed form of this infinite series?
Proving that a process is a Brownian motion
How to prove that $\lim \frac{1}{n} \sqrt{(n+1)(n+2)… 2n} = \frac{4}{e}$
Krull dimension and transcendence degree
Finite groups with periodic cohomology
A characterization of trace class operators

Studying from Roman’s Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ by showing that there is an unique linear transformation

$$\theta:U^* \otimes V^* \to (U \otimes V)^*$$

defined by $\theta(f \otimes g)=f \odot g$ where $(f \odot g)(u \otimes v)=f(u)g(v)$

- How to find $A$ from $Ax=b$?
- Null Space of Transformation
- $A^3+A=0$ We need to show $\mathrm{rank}(A)=2$
- In a complex vector space, $\langle Tx,x \rangle=0 \implies T = 0$
- Are non-degenerate bilinear forms equivalent to isomorphisms $V \cong V^*$?
- Continuous deformations of points in $\mathbb{R}^n$ in a monotonic fashion

I want to prove that $U^* \otimes V^* \cong \hom(U,V;\mathbb{F})$. I tried to use the universal property by fixing $f \in U^*, g\in V^*$ and defining $S: U^*\times V^* \to \hom(U,V;\mathbb{F})$ as $S(f,g)=F_{f,g}$, where $F_{f,g}(u,v)=f(u)g(v)$, but I got stuck.

- Let V be a vector space. If every subspace of V is T-invariant, prove that there exist a scalar multiple c such that T=c1v
- A proof about polynomial division
- There is a $3\times 3 $ orthogonal matrix with all non zero entries.?
- Formalizing the idea of a set $A$ *together* with the operations $+,\cdot$''.
- Operator norm and tensor norms
- General Steinitz exchange lemma
- $\beta_k$ for Conjugate Gradient Method
- Order of $\mathrm{GL}_n(\mathbb F_p)$ for $p$ prime
- Determinant of a Special Symmetric Matrix
- p-norm and relative relations questions

It is much easier to prove that $(U\otimes V)^{\ast}\cong \mathrm{hom}(U,V;\mathbb{F})$. This is because any bilinear map $F:U\times V\to\mathbb{F}$ induces a unique homomorphism $F’:U\otimes V\to\mathbb{F}$ and vice versa. Notice that $F’\in (U\otimes V)^{\ast}$; the universal property gives us the bijection.

- Show group of order $4n + 2$ has a subgroup of index 2.
- Prove $4+\sqrt{5}$ is prime in $\mathbb{Z}$
- Venn diagram for conditional probability property of Independent Events
- Differentiate $\sin \sqrt{x^2+1} $with respect to $x$?
- Name for a certain “product game”
- In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?
- How to show that $m^*(A \cup B) + m^*(A \cap B) \leq m^*(A)+m^*(B)$ for any $A,B \subseteq \mathbb{R}$?
- Uniformization theorem and metrics on Riemann surfaces
- Degree of the splitting field of $x^{p^2} -2$ over $\mathbb{Q}$, for prime p.
- Free groups: unique up to unique isomorphism
- If a cyclotomic integer has (rational) prime norm, is it a prime element?
- How to formulate continuum hypothesis without the axiom of choice?
- Give an example of a nonabelian group in which a product of elements of finite order can have infinite order.
- Proof of convexity of $f(x)=x^2$
- Prove $x_n$ converges IFF $x_n$ is bounded and has at most one limit point