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An inner product on a vector space $V$ over $\mathbb{F}$ (where $\mathbb{F}$ is $\mathbb{R}$ or $\mathbb{C}$) is a function $\langle \cdot , \cdot \rangle : V \times V \rightarrow \mathbb{F}$ that is conjugate symmetric, linear in its first argument, and positive definite. Example: $\langle x, y \rangle = x_1 \overline{y}_1 + ... + x_n \overline{y}_n$ for $x, y \in \mathbb{C}^n$.

A norm on a vector space satisfies: 1) Non-negativity: $\|x\| \geq 0$, 2) Definiteness: $\|x\| = 0$ if and only if $x = 0$, 3) Homogeneity: $\|\lambda x\| = |\lambda| \|x\|$, 4) Triangle Inequality: $\|x + y\| \leq \|x\| + \|y\|$. Example: In $\mathbb{R}^n$, the Euclidean norm satisfies these properties.

For any orthonormal sequence $(e_n)$ in an inner product space, and any vector $x$, the sum of the squares of the inner products is less than or equal to the square of the norm of $x$: $\sum |\langle x, e_n \rangle|^2 \leq \|x\|^2$. Example: In $l^2$, the space of square-summable sequences, taking $e_n$ as standard basis vectors and $x = (1,1,1,...)$ satisfies the inequality.

If $W$ is a subspace of an inner product space $V$, the projection of a vector $v$ onto $W$ is the unique vector $w$ in $W$ such that $v - w$ is orthogonal to every vector in $W$. Example: Projecting $(3,3)$ onto the span of $(1,0)$ in $\mathbb{R}^2$ gives $(3,0)$.

An operator $T$ on a Hilbert space is self-adjoint if $T = T^*$, meaning it equals its own adjoint. Example: The matrix $\begin{pmatrix}2 & 0\\0 & 3\end{pmatrix}$ is self-adjoint on $\mathbb{R}^2$.

In an inner product space, the norm of the sum of two vectors is less than or equal to the sum of their norms: $\|x + y\| \leq \|x\| + \|y\|$. Example: In $\mathbb{R}^2$, for vectors $x = (1,1)$ and $y = (-1,2)$, $\sqrt{10} \leq \sqrt{2} + \sqrt{5}$.

For a linear operator $T$ on a Hilbert space $H$, the adjoint $T^*$ is the unique operator such that $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x, y \in H$. Example: For the matrix $A = \begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}$ over $\mathbb{C}^2$, the adjoint is $A^* = \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$.

The orthogonal complement of a subspace $W$ in an inner product space $V$ is the set of all vectors in $V$ that are orthogonal to every vector in $W$. Example: In $\mathbb{R}^3$, the orthogonal complement of the $x$-$y$ plane is the $z$-axis.

For all vectors $x$ and $y$ in an inner product space, $|\langle x, y \rangle| \leq \|x\| \|y\|$, with equality if and only if $x$ and $y$ are linearly dependent. Example: In $\mathbb{R}^3$, for $x = (1, 0, 1)$ and $y = (0, 1, 1)$, the inequality holds as $0 \leq \sqrt{2}\sqrt{2}$.

Two vectors $x$ and $y$ in an inner product space are orthogonal if $\langle x, y \rangle = 0$. Example: In $\mathbb{R}^2$, the vectors $(1, 0)$ and $(0, 1)$ are orthogonal.

The inner product can be expressed in terms of the norm: for real vector spaces $\langle x, y \rangle = \frac{1}{4}(\|x+y\|^2 - \|x-y\|^2)$, and for complex vector spaces $\langle x, y \rangle = \frac{1}{4}(\|x+y\|^2 - \|x-y\|^2 + i\|x+iy\|^2 - i\|x-iy\|^2)$. Example: In $\mathbb{R}^2$, $\langle (1, 0), (0, 1) \rangle = 0$ can be computed using the identity.

For an orthonormal basis $(e_n)$ of a Hilbert space $H$ and any vector $x \in H$, the sum of the squares of the inner products equals the square of the norm of $x$: $\sum |\langle x, e_n \rangle|^2 = \|x\|^2$. Example: In $\mathbb{C}^n$ with the standard basis, for vector $x$, Parseval's identity confirms that the norm squared is equal to the sum of the squares of the modulus of $x$'s components.

A Hilbert space is a complete inner product space. These spaces are generalizations of Euclidean spaces to possibly infinite dimensions. Example: $L^2([a, b])$, the space of square-integrable functions on the interval $[a, b]$, is a Hilbert space.

A method for obtaining an orthonormal set from a linearly independent set. It proceeds by orthogonalizing the vectors with respect to one another and then normalizing them. Example: Starting with $(1,0,0)$ and $(1,1,0)$ in $\mathbb{R}^3$, the normalized orthogonal vectors would be $(1,0,0)$ and $(0,1,0)$.

A linear operator $U$ on a Hilbert space $H$ is unitary if $U^*U = UU^* = I$, where $I$ is the identity operator. Example: The matrix $\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}$ is unitary on $\mathbb{C}^2$.

In an inner product space, the relationship $\|x + y\|^2 + \|x - y\|^2 = 2(\|x\|^2 + \|y\|^2)$ holds. Example: For vectors $x = (1,0)$ and $y = (0,1)$ in $\mathbb{R}^2$, $2^2 + 2^2 = 2(1^2 + 1^2)$.

A principle stating that an analytic function defined on the upper half-plane that takes real values on the real axis can be extended to the lower half-plane by reflection. Example: If $f(z)$ is analytic in the upper half-plane and real on the real axis, then $f(\overline{z}) = \overline{f(z)}$ extends $f$ to the lower half-plane.

Every continuous linear functional $f$ on a Hilbert space $H$ can be represented as an inner product with a unique vector $y$ in $H$: $f(x) = \langle x, y \rangle$ for all $x \in H$. Example: In $l^2$, the space of square-summable sequences, the functional $f((x_n)) = x_1$ is represented by the sequence $(1, 0, 0, ...)$.

Every compact self-adjoint operator on a Hilbert space has an orthonormal basis of eigenvectors. Example: The matrix $\begin{pmatrix}2 & 0\\0 & 3\end{pmatrix}$ represents a self-adjoint operator on $\mathbb{R}^2$ with eigenvectors $(1, 0)$ and $(0, 1)$.