Assuming that $(f_n)_n$ is bounded, does $\sup_{n\in\mathbb N}||f_n||_H <+\infty$ hold?

Question

Let $(H, \|\cdot\|)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$.

I was wondering if this information is enough to conclude that $$\sup_{n\in\mathbb N}\|f_n\|_H <+\infty.$$

I'd say that the answer is yes because being bounded in $H$ means that there exists $R>0$ such that $\|f_n\|_H \le R$. With this information, passing to the supremum, one should have $$\sup_{n\in\mathbb N}\|f_n\|_H \le R <+\infty.$$

Anyone can help me in understanding if I argue it correctly?

Answer 1

To provide an answer for this question, yes, what you have done is correct.

In fact, the converse is also true, in the sense that if $(f_{n})_{n\in\mathbb{N}}$ is a sequence in a Hilbert space $H$ and if

$$\sup_{n\in\mathbb{N}} \|f_{n}\| < +\infty ,$$

it follows that $(f_{n})_{n\in\mathbb{N}}$ is a bounded sequence. This is because $\|f_{n}\| \leq \sup_{m\in\mathbb{N}} \|f_{m}\|$ for all $n\in\mathbb{N}$.

Furthermore, it is enough for $H$ to simply be a normed space, because all you needed to conclude the statement was a norm and the definition of a bounded subset of a norm.