Properties of Injective Operator on Hilbert Space

Question

I am new to functional analysis and have the following issue:

Given an infinite dimensional Hilbert space $H$ and an operator $f: H \times \Omega \to H$, where $\Omega$ is some finite dimensional vector space, how can one determine if $f$ is injective? That is, is it possible that such an operator could be one-to-one?

Any good references would be much appreciated.

Answer 1

Let $f: \mathbb{R}^n \times l_2 (\mathbb{R} ) \to l_2 $ $$ f((x_1 ,x_2 , ..., x_n ) , (y_i )_{n\geq 1} ) = (x_1 ,x_2 , ..., x_n , y_1 , y_2 , ...)$$ then $f$ is injective.