I am studying Linear Algebra II, and I came across several questions in which, for a certain linear transformation ($T\colon\mathbf{V}\to\mathbf{V}$) I was told that: $||T(a)|| \leq ||a||.$
I am not completely certain how to use this information. For instance, consider the following question (please forgive my translation, it's the first time I write math in English):
For a linear transformation $T\colon\mathbf{V}\to\mathbf{V}$in a unitary space [i.e., complex inner product space], such that
- $|c|=1$ for every eigenvalue $c$ of $T$;
- $||T(a)|| \leq ||a||$ for every vector $a$ in $\mathbf{V}$; prove that T is a unitary operator.
How does the fact that $||T(a)|| \leq ||a||$ help me?
Thanks.