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(1)Define $$T: U\rightarrow \prod V_i,\ T(x)=(T_1(x),\cdots , T_n(x) )$$ Hence we have an existence. (2) Note that any vector in $\prod V_i$ has the form $$ v=(v_1,\cdots, v_n)$$ where $v_i\in V_i$. Hence for all $i$, if $0=p_i(v)$, then $v=(0,\cdots, 0)$. (3) If $S$ is another then $$ p_i\circ (S-T)(x)= 0$$ Hence $(S-T)(x)=0$. Hence we have ...


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The stupid answer is that $\operatorname{Hom}_{\text{cts}}(\mathbb Z, S^1) = \operatorname{Hom}_{\mathbb Z}(\mathbb Z,S^1) = S^1$, so we are done by duality. Here, $S^1$ denotes the unit circle $S^1 \cong \frac{\mathbb R}{2\pi\mathbb Z} \cong \mathbb R/\mathbb Z$, a.k.a. $\mathbb T$. The real answer is that all characters of $S^1$ are of the form $z ...


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Yes, it is. Let $M=\{x\in X: y^*(x)=0 \text{ for all }y^*\in Y^*\}$. This is sometimes denoted ${}^\perp Y^*$ and called the pre-annihilator of $Y^*$. Every $y^*\in Y^*$ induces a linear functional on $X/M$ in a natural way: $y^*(x+M)=y^*(x)$ is well-defined. Conversely, if $\phi$ is a linear functional on $X/M$, then its composition with the projection ...


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If I understand correctly, your question is trying to interpret the dual of your original problem which involves minimizing the cost of goods. You want to minimize cost, while trying to meet all your demand constraints. Then logically, the dual would be maximizing the amount of goods sent out (the demand met?), subject to cost constraints.


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The answer is yes. This follows from the Hahn-Banach theorem. Indeed, for each non-zero element $x$ in a Banach space $X$, there is a norm-one functional $f\in X^*$ such that $\langle f,x\rangle = \|x\|$. To see this, consider the one-dimensional subspace spanned by $x$. Let $\langle f,cx\rangle = c\|x\|$ ($c$ is a scalar). Then $f$ is a norm-one ...


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Assume you have a linear program of the form $\quad$ maximize $\mathbf{n}^T\mathbf{x}$ subject to $A\mathbf{x}\leq\mathbf{a}$. The dual lp is $\quad$ minimize $\mathbf{a}^T\mathbf{u}$ subject to $A^T\mathbf{u}=\mathbf{n}, \mathbf{u}\geq \mathbf{0}$. By weak duality for any solution $\mathbf{x}_0$ of the primal lp and $\mathbf{u}_0$ of the dual lp it ...



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