Infinite dimensional function space known as a Hilbert space, denoted as $H$, and initialized with inner product $(\textbf u|\textbf v)$ is a space in which an operator $L$ (positive and summetric) acts. Another Hilbert space $H_L$ is defined with the inner product being $(L\textbf u|\textbf v)$. It is claimed that if the operator $L$ is positive definite over $H$ then $H_L$ is a subspace of $H$! A proof of this postulate stems form the definition of positive definite operator: $L$ is positive definite if there is a positive number $\gamma$ such that $(L\textbf u|\textbf u)\geq \gamma(\textbf u|\textbf u)$ for all $\textbf u$ in $H$. Then the following inequality pops out: $||\textbf u||^2=(\textbf u|\textbf u)\leq \frac{(L\textbf u|\textbf u)}{\gamma}=\frac {||\textbf u||_L^2}{\gamma}$ and it is stated that namely due to this inequality $H_L$ is a subspace of $H$!
So my question is why the last inequality means that $H_L$ is a subspace of $H$?
Also if $L$ is only positive but not positive definite then why we can't say that $H_L$ is a subspace of $H$?
Actually at the end it was written $H_L\subset H$ which I interpret as $H_L$ is a subspace of $H$ since earlier in the book it was written explicitly that $H_L$ is a space.