Help understanding an intermediate result

Apr 2015
62
6
Sweden
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
 
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