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+# Topic 3: Separable Hilbert spaces
+
+## Infinite-dimensional Hilbert spaces
+
+Recall from Topic 1.
+
+[$L^2$ space](https://notenextra.trance-0.com/Math401/Math401_T1#section-3-further-definitions-in-measure-theory-and-integration)
+
+Let $\lambda$ be a measure on $\mathbb{R}$, or any other field you are interested in.
+
+A function is square integrable if
+
+$$
+\int_\mathbb{R} |f(x)|^2 d\lambda(x)<\infty
+$$
+
+### $L^2$ space and general Hilbert spaces
+
+#### Definition of $L^2(\mathbb{R},\lambda)$
+
+The space $L^2(\mathbb{R},\lambda)$ is the space of all square integrable, measurable functions on $\mathbb{R}$ with respect to the measure $\lambda$ (The Lebesgue measure).
+
+The Hermitian inner product is defined by
+
+$$
+\langle f,g\rangle=\int_\mathbb{R} \overline{f(x)}g(x) d\lambda(x)
+$$
+
+The norm is defined by
+
+$$
+\|f\|=\sqrt{\int_\mathbb{R} |f(x)|^2 d\lambda(x)}
+$$
+
+The space $L^2(\mathbb{R},\lambda)$ is complete.
+
+[Proof ignored here]
+
+> Recall the definition of [complete metric space](https://notenextra.trance-0.com/Math4111/Math4111_L17#definition-312).
+
+The inner product space $L^2(\mathbb{R},\lambda)$ is complete.
+
+#### Definition of general Hilbert space
+
+A Hilbert space is a complete inner product space.
+
+#### General Pythagorean theorem
+
+Let $u_1,u_2,\cdots,u_N$ be an orthonormal set in an inner product space $\mathscr{V}$ (may not be complete). Then for all $v\in \mathscr{V}$,
+
+$$
+\|v\|^2=\sum_{i=1}^N |\langle v,u_i\rangle|^2+\left\|v-\sum_{i=1}^N \langle v,u_i\rangle u_i\right\|^2
+$$
+
+[Proof ignored here]
+
+#### Bessel's inequality
+
+Let $u_1,u_2,\cdots,u_N$ be an orthonormal set in an inner product space $\mathscr{V}$ (may not be complete). Then for all $v\in \mathscr{V}$,
+
+$$
+\sum_{i=1}^N |\langle v,u_i\rangle|^2\leq \|v\|^2
+$$
+
+Immediate from the general Pythagorean theorem.
+
+### Orthonormal bases
+
+#### Definition of orthonormal basis
+
+An orthonormal basis of a Hilbert space $\mathscr{H}$ is a set of orthonormal vectors that spans $\mathscr{H}$.