is bounded linear operator necessarily continuous?

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Solution 1

An operator $C$ is bounded iff the set {$\|Cx\|:\|x\|\leq 1$} is bounded $\Leftrightarrow$ there is a $M<\infty:\|Cx\|\leq M\|x\|$ for every $x\in U$.

Let $ε>0$. If $x,y\in U:\|x-y\|<ε/M$, then $\|Cx-Cy\|\leq M\|x-y\|<ε$. Thus $C$ is not only continuous but uniformly continuous also.

So, a bounded operator is always continuous on norm-spaces. Banach space is a norm-space which is complete, thus things are not different there.

Solution 2

This property is unrelated to the completeness of the domain or range, but instead only to the linear nature of the operator. Yes, a linear operator (between normed spaces) is bounded if and only if it is continuous.

Added @Dimitris's answer prompted me to mention, beyond the fact that the implication on normed spaces indeed is an equivalence, that it's the converse which holds in the wider context of topological vector spaces, while the proposition mentioned here fails: there are bounded discontinuous linear operators, yet every continuous operator remains bounded.

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Updated on August 01, 2022

Comments

  • user106357
    user106357 over 1 year

    Let $U, V$ be separable Banach spaces.

    Suppose we have a bounded, linear operator $C : U\to V$.

    Questions are the following

    *) Shall $C$ be continuous since $V$ is a Banach space?

    *) In general, is a bounded linear operator necessarily continuous (I guess the answer is no, but what would be a counter example?)

    *) Are things in Banach spaces always continuous?

    • Admin
      Admin about 10 years
      Bounded linear operators are continuous. (Think about how Lipschitz condition implies uniform continuity for functions on real line). Things in Banach spaces aren't always continuous though.
    • Noix07
      Noix07 almost 8 years
      actually why don't we directly learn: "bounded linear operators" equivalent to "Lipschitz continuous one", but only equivalent to continuous ones?
    • MSIS
      MSIS almost 3 years
      @user27126: Is every continuous operator bounded? I know the answer is yes in finite-dimensional spaces, but how about in infinite-dimensional ones? I am thinking maybe, e.g., the differentiation operator which I know is unbounded is also continuous?
    • Taleofwoe
      Taleofwoe almost 2 years
      Differentiation is not necessarily continuous, consider the space of smooth 2 $\pi$ periodic functions and the sequence sin(nx)/x and the sup-norm.
  • user106357
    user106357 about 10 years
    Thanks Jonathan and Sanchez.In this case, so instead of saying "a bounded linear operator $C: U\to V$", one can say $C\in\mathcal{L}(U,V)$, where $\mathcal{L}(U,V)=\{C: U\to V\,\,|\,\, C\,\,\text{is linear and continuous}\}$?
  • Jonathan Y.
    Jonathan Y. about 10 years
    You may, only I've only seen this notation used as a synonym of $\mathrm{Hom}(U,V)$, the space of all linear maps between them. Perhaps another notation would serve you better.
  • user106357
    user106357 about 10 years
    $\text{Hom}(X,Y)=\{C: U\to V\,\,|\,C\,\,\text{is linear}\}$, which means $C$ is not necessarily continuous. In finite dimensions, the two notations are the same. But this is not true in infinite dimensions. Correct me if I'm wrong
  • Jonathan Y.
    Jonathan Y. about 10 years
    You're correct. I was saying I've only seen $\mathcal{L}(U,V)$ being used as an alternate notation to $\mathrm{Hom}(U,V)$.
  • Stromael
    Stromael about 10 years
    The standard notation that I have encountered is $B(U,V)$, meaning the space of bounded linear operators that map the entirety of the linear space $U$ into some/all of the linear space $V$. This is identical to the space of continuous linear operators $U\rightarrow V$. When $U=V$ then it is simply notated $B(V)$.