# Help understanding an intermediate result

#### Krisly

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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