I can find no precise definitions on the internet for the $L^2$ and $\ell^2$ norms. Certain websites keep switching between the two. Can someone please help me?
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Regarding the switching I would like to add that $L^2([0,1])$ and $\ell^2$ are isomorphic as Hilbert spaces (as they are both separable and infinite-dimensional). That means: If $(f_n)_{n\in\mathbb N}$ is an orthonormal basis of $L^2([0,1])$, for example the basis $\{\exp(2\pi i n \,\cdot\,) : n \in \mathbb Z\}$, then \begin{align*} L^2([0,1]) &\to \ell^2\\ f &\mapsto (\left<f, e_n\right>)_n \end{align*} is an isometric isomorphism of Hilbert spaces, so especially $\|f\|_{L^2} = \|(\left<f, e_n\right>)_n\|_{\ell^2}$. |
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The scalar product on $L^2$ is given by $\langle f,g\rangle=\int_X \bar{f}{g} \ d\mu$, whereas the scalar product on $\ell^2$ is given by $\langle x,y\rangle =\sum_{i \in \mathbb{N}} \bar{x_i} y_i$. In both cases the norm is given as usual in Hilbert spaces by $\lVert f\lVert=\sqrt{\langle f,f\rangle}$. So $\ell^2$ is a special case of $L^2$ with $X=\mathbb{N}$ and the counting measure $\mu$. |
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