linear operators on finite dimensional normed vector space is bounded Let $H$ be a finite dimensional Hilbert space. Let $T:H\to H$ be a linear operator. Show that $T$ is bounded.
my work: Since $H$ is finite dimensional, there exists an orthonormal basis $\{e_1,...,e_n\}$. I want to show that if $v \in H$ and $\|v\|=1$, then $\langle Tv,v \rangle$ is bounded. Let $v=c_1e_1+...+c_ne_n$, then $$\langle Tv,v\rangle=\big\langle c_1T(e_1)+c_2T(e_2)+...+c_nT(e_n),c_1e_1+...+c_ne_n\big\rangle$$
Can anyone tell me how to proceed or is there any better way to prove this?
 A: Consider a basis of $H$, say $\{e_i\}$.  Then, note that for any $x = \sum_{i=1}^n x_ie_i$ (where $n = \dim H$) we have $$||Tx||^2 = \langle Tx,Tx\rangle = \left\langle  \sum_{i=1}^n x_i T(e_i),\sum_{i=1}^n x_i T(e_i) \right\rangle = \sum_{i,j=1}^nx_i \overline{x_j}\left\langle T(e_i),T(e_j)\right\rangle$$
Now, let $M = \displaystyle\max_{1 \leq i,j \leq n} \langle T(e_i),T(e_j)\rangle$, existing because we are taking the maximum over a finite set. Then, we have:
$$
||Tx||^2 = \sum_{i,j=1}^n x_i \overline{x_j}\langle T(e_i)T(e_j)\rangle \leq M \sum_{i,j=1}^n x_i \overline{x_j} \leq M ||x||^2
$$
taking square roots on both sides tells us that the norm of $T$ exists, and is less than or equal to $\sqrt M$.
A: Notice that $$\|v\|^2 = \left\|\sum_{i=1}^n c_ie_i\right\|^2 = \sum_{i=1}^n |c_i|^2$$
Thus for $\|v\| = 1$ we have:
\begin{align}\left|\langle Tv, v\rangle\right| &= \left|\left\langle\sum_{i=1}^n c_iTe_i, \sum_{j=1}^n c_je_j \right\rangle\right| \\
&= \left|\left\langle\sum_{i=1}^n c_i\left(\sum_{j=1}^n \langle Te_i, e_j\rangle e_j\right), \sum_{i=1}^n c_ie_i \right\rangle\right| \\
&= \left|\left\langle\sum_{j=1}^n \left(\sum_{i=1}^n c_i\langle Te_i, e_j\rangle\right)e_j, \sum_{i=1}^n c_ie_i \right\rangle\right| \\
&= \left|\sum_{j=1}^n \overline{c_j}\left(\sum_{i=1}^n c_i\langle Te_i, e_j\rangle \right)\right| \\
&\le \sum_{j=1}^n |c_j|\left(\sum_{i=1}^n |c_i||\langle Te_i, e_j\rangle| \right) \\
&\le \max\limits_{1 \le i \le j \le n}|\langle Te_i, e_j\rangle| \cdot \sum_{j=1}^n |c_j|\left(\sum_{i=1}^n |c_i| \right) \\
&= \max\limits_{1 \le i \le j \le n}|\langle Te_i, e_j\rangle| \cdot \left(\sum_{j=1}^n |c_j|\right)^2 \\
&\stackrel{CSB}{\le} \max\limits_{1 \le i \le j \le n}|\langle Te_i, e_j\rangle| \cdot \sqrt{\sum_{i=1}^n |c_i|^2}\sqrt{\sum_{i=1}^n 1^2}  \\
&\le \max\limits_{1 \le i \le j \le n}|\langle Te_i, e_j\rangle| \cdot \sqrt{n} \cdot \|v\|\\
&\le \max\limits_{1 \le i \le j \le n}|\langle Te_i, e_j\rangle| \cdot \sqrt{n}
\end{align}
Therefore, the set
$$\big\{\left|\langle Tv, v\rangle\right| : v \in H, \|v\| = 1\big\}$$ is bounded.
