Suppose that $T \in L(V,W)$. Prove that if Img$(T)$ is dense in $W$ then $T^*$ is one-to-one.
Answer
Suppose $T^*$ is not one-to-one. Then there exists $f_1,f_2 \in W^*$ with $f_1\neq f_2$ such that \[ T^*(f_1)=T^*(f_2). \] Thus \[ f_1(Tv)=f_2(Tv), \forall v\in V. \] and hence $(f_1-f_2)(Tv)=0$, $\forall v \in V$, i.e. \[f_1-f_2=0, \forall w\in Img (T).\] Since $f_1-f_2$ is continuous, it also vanishes on $\overline{Img(T)}=W$, and therefore $f_1-f_2=0$ on $W$, which contradicts the assumption $f_1 \neq f_2$. The proof is complete.
Savionf
574
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 3110 views
- $18.00
Related Questions
- Constructing Monotonic Sequences Converging to an Accumulation Point in a Subset of $\mathbb{R}$
- real analysis
- Assume there is no $x ∈ R$ such that $f(x) = f'(x) = 0$. Show that $$S =\{x: 0≤x≤1,f(x)=0\}$$ is finite.
- Prove that $\frac{d \lambda}{d \mu} = \frac{d \lambda}{d \nu} \frac{d \nu}{d \mu}$ for $\sigma$-finite measures $\mu,\nu, \lambda$.
- Does the sequence $f_n=\arctan (\frac{2x}{x^2+n^3})$ converge uniformly on $\mathbb{R}$?
- Prove that $S \subseteq X$ is nowhere dense iff $X-\overline{S}$ is dense.
- [Real Analysis] Show that $B$ is countable.
- Rank, Range, Critical Values, Preimage, and Integral of Differential Forms
