# Thread: Help understanding an intermediate result

1. ## Help understanding an intermediate result

Hello everyone

I am reading a section of a textbook concerning The Galerkin Method, the setting are a separable Hilbert space with a Shauder basis $\displaystyle \left(\phi_i\right)_{i\in \mathbb{N}}$ and a matrix $\displaystyle A_n = \left(a\left(\phi_i, \phi_j \right)\right) \: 1 \leq i,j \leq n$ the Author then arrives at the following result,
... there exist some constant c > 0 such that

$\displaystyle \forall \lambda \in \mathbb{R}^n \: c \left| \lambda \right| \leq \left| \left| \sum_i^n \lambda_i \phi_i \right| \right|$

Hence

$\displaystyle \forall \lambda \in \mathbb{R}^n \: \langle A_n \lambda , \lambda \rangle \geq \alpha c^2 \left| \lambda \right| ^2$

From the above, it follows that $\displaystyle A_n$ is one to one (that is, $\displaystyle ker(A_n)= \left \lbrace 0 \right \rbrace$ )
My question is then why does it follow (or how does it follow) that the matrix $\displaystyle A_n$ is injective?

Thanks

P.S $\displaystyle \langle \: \cdot \: , \: \cdot \: \rangle$ is the Euclidean scalar product and $\displaystyle \left| \: \cdot \: \right|$ is the Euclidean norm