Norm of Lower Traingular matrix of a LU decomposition $\|L\|_{\infty}\le n $

Question

In LU factorization of a square matrix , show that

1. $\|L\|_F \le \frac{n(n+1)}{2}$

2. $\|L\|_{\infty}\le n $

3. $\|L\|_1\le n$

4. $\|L\|_2 \le n$

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Now here in $L$ all diagonal elements are $1$.Now Frobenius norm is $\sqrt{trace(L^tL)}$ . Therefore we have to prove that $\sqrt{trace(L^tL)} \le \frac{n(n+1)}{2}$ .

For (2) $ \|L\|_{\infty} =max_i\sum_{j=1}^n|a_{ij}| \le n$ then sum of all row sum will be $\le$ n but how if some elements is bigger than n??