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I want to prove that a bounded linear functional $f$, must be continuous.

I have defined: $f$ is bounded means that $\exists c> 0, |f(x)|\leq c\|x\|, \quad \forall x\in X$ and continuous means that $x_n\to x \implies f(x_n)\to f(x)$.

Proof: Let $f$ be bounded. Then $\exists c, |f(x)| \leq c\|x\|$

Then

\begin{align} &\lim_{n\to\infty} |f(x_n-x)|\leq \lim_{n\to\infty} c\|x_n-x\|\\ \implies& \lim_{n\to\infty} |f(x_n)-f(x)|\leq c\|x-x\|=0\\ \implies& \lim_{n\to\infty} f(x_n)=f(x) \end{align}

Is that all there is to it?

I used linearity in the second line, but what if we removed the condition that $f$ is linear. Is a non-linear bounded functional $f$ necessarily continuous

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(1) Yes, that's all.

(2) Without linearity, the statement becomes wrong, even in the $X = \mathbf R$ case. Consider for example, $f \colon \def\R{\mathbf R}\R\to \R$ given by $$ f(x) = \begin{cases} x & x \in [-1,1] \\ 0 & x \not\in [-1,1] \end{cases} $$ Then $f$ is bounded, as $|f(x)| \le |x|$ for all $x \in \R$, but not continuous.

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