If $V$ and $W$ are vector spaces, $\beta=\{v_1, \ldots , v_n\}$ is a finite a basis for $V$ and $\{w_1, \ldots , w_n\}\subset W$, we know there is an unique linear transformation $T:V\rightarrow W$ such that $T(v_i)=w_i$ for $i=1, 2, \ldots , n$ Is this valid when $V$ is not finite-dimensional?

Define $G = \Bbb{Q}_{\gt 0}$ to be the (multiplicative) group of positive rational numbers. Suppose that there is a group surjection: $$ f : G \twoheadrightarrow \Bbb{Z} $$ and we took its kernel $N = \ker f$. Then of course $G/N \simeq \Bbb{Z}$ by first isomorphism theorem. By the third isom...

Add two rational numbers... esoterically Objective Build a binary operator \$\star : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}\$ such that, there exists a bijection \$i : \mathbb{Z} \to \mathbb{Q}\$ such that, for every integer \$M\$ and \$N\$, \$M \star N = i^{-1}(i(M) + i(N))\$ holds. Or in m...

Let $f:\Bbb [a,b]\to \Bbb R$ satisfy the following : (i) $f(x)$ is derivable in $[a,b]$ (ii) $f'(a)=\alpha\neq f'(b)=\beta$ and (iii) $\exists \gamma\in (\alpha,\beta)$ then, there exists at least one value of $x$ say $\xi$ between $a$ and $b$ such that $f'(\xi)= \gamma.$ I tried to prove the abo...

Let me take the set $A = \{1, 2\}$ as the alphabet. By the bijective binary numeral system, $A^*$ has one-to-one correspondence to the set of nonnegative integers $\mathbb{N}$. As such, each language $L \subset A^*$ corresponds to a subset of $\mathbb{N}$. Now consider the power set of $\mathbb{N...

Let $f:\Bbb [a,b]\to \Bbb R$ satisfy the following : (i) $f(x)$ is continuous in $[a,b]$ (ii) $f(x)$ is derivable in $(a,b)$ (iii) $f(a)=f(b)$ then, $\exists c\in (a,b)$ such that $f'(c)= 0.$ I tried to prove this theorem as follows: Since, $f(x)$ is continuous in $[a,b]$ so by Maximum-Minimum Th...