# The Gram-Schmidt process: on the equality of the generated subspace.

We work in an infinite Hilbert space $$H$$.

Let $$(x_k)$$ an arbitrary linearly independent sequence, then we define $$e_1=\frac{x_1}{\lVert x_1\Vert}\quad\text{and}\quad e_k=\frac{x_k-\sum_{j For construction $$(e_k)$$ is an ortonormal sequence.

I must prove that $$\text{span}(e_1, \dots, e_k)=\text{span}(x_1,\dots, x_k)\quad\forall k\ge 1\tag 1.$$

First, for indution on $$k$$ we prove that $$(e_1,\dots, e_k)\subseteq \text{span}(x_1,\dots, x_k)\tag 2.$$ For $$k=1$$ we have that $$(e_1)\subseteq\text{span}(x_1)$$ for the definition of $$e_1$$. Suppose that $$(e_1,\dots, e_k)\subseteq \text{span}(x_1,\dots, x_k)$$ is true for same $$k$$ and we prove that $$(e_1,\dots, e_{k+1})\subseteq \text{span}(x_1,\dots, x_{k+1}).$$

At this point I have problems, because I don't know how to do it after doing the following:

$$\{e_1,\dots, e_{k+1}\}=\{e_1, \dots, e_k\}\cup \{e_{k+1}\}\subseteq\text{span}(x_1,\dots, x_k)\cup \{e_{k+1}\}.$$

From $$(2)$$ follow that $$\text{span}(e_1,\dots, e_k)\subseteq \text{span}(x_1,\dots, x_k).$$

On the other hand the vectors $$e_1, \dots, e_k$$ are linearly independent since they are ortogonal. Therefore $$\dim\text{span}(e_1,\dots, e_k)=k=\dim\text{span}(x_1,\dots, x_k),$$ from which follows $$(1)$$.

Could you help me complete the proof of $$(2)$$?

For your inductive step, you have that $$e_{k+1}=\beta_{k+1}x_{k+1}-\sum_{j=1}^k\beta_je_j$$ for certain coefficients $$\beta_1,\ldots,\beta_{k+1}$$. The inductive hypothesis gives you that $$\sum_{j=1}^k\beta_je_j=\sum_{j=1}^k\gamma_jx_j$$ for certain coefficients $$\gamma_1,\ldots,\gamma_k$$. Then $$e_{k+1}=\beta_{k+1}x_{k+1}-\sum_{j=1}^k\gamma_jx_j\in\operatorname{span}\{x_1,\ldots,x_{k+1}\}.$$ That completes the induction.