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----
-title: Inner Product Spaces
-parent: Functional Analysis Basics
-nav_order: 1
----
-
-# {{ page.title }}
-
-{% definition Inner Product Space %}
-An *inner product* on a real or complex vector space $X$
-is a mapping
-
-$$
-\innerp{\cdot}{\cdot} : X \times X \to \KK
-$$
-
-that is
-
-- linear in its second argument
-- conjugate symmetric
-- nondegenerate
-
-An *inner product space* is a pair $(X,\innerp{\cdot}{\cdot})$
-consisting of a real or complex vector space $X$
-and an inner product $\innerp{\cdot}{\cdot}$ on $X$.
-{% enddefinition %}
-
-{% proposition Norm Induced by an Inner Product %}
-If $\innerp{\cdot}{\cdot}$ is an inner product
-on a real or complex vector space $X$, then
-
-$$
-\norm{x} = \sqrt{\innerp{x}{x}} \qquad \forall x \in X
-$$
-
-defines a norm on $X$.
-{% endproposition %}
-
-In this sense, every inner product space is also a normed space.
-As a consequence it is also a metric space and a topolgical space.
-
-The next theorem shows how the inner product can be recovered from the norm.
-
-{% theorem * Polarization Identity %}
-For all vectors $x$ and $y$ of a real inner product space
-
-$$
-4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2.
-$$
-
-For all vectors $x$ and $y$ of a complex inner product space
-
-$$
-4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2 + i \norm{x-iy}^2 - i \norm{x+iy}^2.
-$$
-{% endtheorem %}
-
-Note that the complex polarization identity takes the slightly different form
-
-$$
-4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2 + i \norm{x\mathrel{\color{red}+}iy}^2 - i \norm{x\mathrel{\color{red}-}iy}^2,
-$$
-
-if we follow the convention that the inner product is conjugate linear in its second argument.
-
-{% proof %}
-In the real case, the inner product is symmetric, and we have
-
-$$
-\norm{x \pm y}^2 = \norm{x}^2 \pm 2 \innerp{x}{y} + \norm{y}^2
-$$
-
-for all vectors $x$ and $y$.
-Taking the difference yields the desired result.
-
-In the complex case, the inner product is conjugate symmetric, and we have
-
-$$
-\norm{x \pm y}^2 = \norm{x}^2 \pm 2 \Re \innerp{x}{y} + \norm{y}^2
-$$
-
-for all vectors $x$ and $y$. This implies
-
-$$
-\begin{aligned}
-\norm{x + \phantom{i}y}^2 - \norm{x - \phantom{i}y}^2 &= 4 \Re \innerp{x}{y}, \\
-\norm{x - iy}^2 - \norm{x + iy}^2 &= 4 \Im \innerp{x}{y}.
-\end{aligned}
-$$
-
-The second equation follows from the first by
-substituting $y$ with $-iy$ and
-using that $\Re \innerp{x}{-iy} = \Re (-i\innerp{x}{y}) = \Im \innerp{x}{y}$.
-To obtain the polarization Identity, multiply the second equation with $i$ and then add it to the first.
-{% endproof %}
-
-{% theorem * General Polarization Identity %}
-Let $X$ be a complex inner product space.
-Let $\zeta$ be a $n$-th root of unity with $\zeta \ne 1$ and $\zeta^2 \ne 1$.
-Then
-
-$$
-\innerp{x}{y} = \frac{1}{n} \sum_{k=0}^{n-1} \zeta^k \norm{x + \zeta^k y}^2 \qquad \forall x,y \in X.
-$$
-{% endtheorem %}
-
-As a special case, for $\zeta = i$ and $n=4$, we obtain
-
-$$
-\innerp{x}{y} = \frac{1}{4} \sum_{k=0}^{3} i^k \norm{x + i^k y}^2.
-$$
-
-{% proof %}
-TODO
-{% endproof %}
-
-For an arbitrary normed space,
-the polarization identity does not, in general,
-define an inner product.
-The following theorem, gives a condition for when it does.
-
-{% theorem * Parallelogram Law %}
-Let $X$ be a real or complex normed space.
-A norm $\norm{\cdot}$ on $X$ is induced by
-an inner product $\innerp{\cdot}{\cdot}$ on $X$,
-if and only if $\norm{\cdot}$ satisfies the *parallelogram law*
-
-$$
-\norm{x+y}^2 + \norm{x-y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2 \qquad \forall x,y \in X.
-$$
-
-In this case, the inner product is uniquely determined by $\norm{\cdot}$ and given by the polarization identity.
-{% endtheorem %}
-
-{% theorem * Cauchy–Schwarz Inequality %}
-For all vectors $x$ and $y$ of an inner product space (with inner product $\innerp{\cdot}{\cdot}$ and induced norm $\norm{\cdot}$)
-
-$$
-\abs{\innerp{x}{y}} \le \norm{x} \norm{y},
-$$
-
-and equality holds precisely when $x$ and $y$ are linearly dependent.
-{% endtheorem %}
-
-Expressed only in terms of the inner product, the Cauchy–Schwarz Inequality reads
-
-$$
-\abs{\innerp{x}{y}}^2 \le \innerp{x}{x} \innerp{y}{y}.
-$$
-
-{% proof %}
-TODO
-{% endproof %}
-
-{% corollary Continuity of the Inner Product %}
-The inner product is jointly norm continous.
-{% endcorollary %}
-
-## Orthogonality
-
-{% definition Orthogonal Vectors %}
-Two vectors $x$ and $y$ of an inner product space $(X,\innerp{\cdot}{\cdot})$
-are said to be *orthogonal* or *perpendicular* if $\innerp{x}{y} = 0$,
-and this fact is indicated by writing $x \perp y$.
-{% enddefinition %}
-
-{% theorem * Pythagoras’ Theorem %}
-For all vectors $x$ and $y$ of an inner product space we have
-
-$$
-x \perp y \iff \norm{x+y}^2 = \norm{x}^2 + \norm{y}^2.
-$$
-{% endtheorem %}
-
-{% proof %}
-Immediate.
-{% endproof %}