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Poland and the 2022 Russian invasion of Ukraine

Background
Geography


 * North European Plain, European Plain
 * Carpathian mountains
 * Kaliningrad, Poland–Russia border, Lithuania–Poland border, Suwałki Gap

Polish international relations


 * Polish-Ukrainian relations (pre-invasion)
 * Polish-Russian relations (pre-invasion)
 * Polish-Belarusian relations (pre-invasion)
 * 2021–2022 Belarus–European Union border crisis
 * Polish-Hungarian relations (pre-invasion)

Prelude to the 2022 invasion

 * Polish President goes to Winter Olympics in China to talk to Xi Jinping
 * List of projects of the Belt and Road Initiative

Poland during the first stage of the invasion

 * Visit of Polish, Czech, and Slovenian heads of states with Volodymyr Zelenskyy in Kiev during the Kyiv offensive (2022).

Poland, Ukraine, Russia energy/oil

 * Poland's preparations for becoming independent of Russian natural gas. Infrastructure such as pipelines and LNG terminals.
 * Russia stops supplying natural gas to Poland (and Bulgaria).


 * Ukraine's sources of energy before the invasion.
 * How these sources were cut off by the Russian naval blockade.
 * Post-invasion energy shortages in Ukraine.
 * Infeasibility of transferring oil to West Ukraine from Romania, Hungary, or Slovakia due to the Carpathian mountains.
 * Sudden dependence on Poland for energy supply, and the limited pre-invasion infrastructure for transferring oil and energy from Poland to Ukraine.

Military aid to Ukraine

 * Hungary not allowing military aid to Ukraine to pass through Hungarian territory.
 * Strain in Polish-Hungarian relations.
 * Polish military aid to Ukraine:
 * Piorun (missile)
 * Proposed transfer of Polish MiG-29 fighter jets to Ukraine.
 * 240+ Polish T-72 tanks sent to Ukraine. Lack of replacements.

Changes to the Polish armed forces

 * Increase in Polish active-duty personnel from 113,000 to 250,000.
 * Increase in Polish territorial personnel from 30,000 to 50,000.
 * Increase in Polish defense spending from 2% to 3%.
 * Poland buying U.S. military equipment.

Transit of non-Polish military aid to Ukraine through Polish territory

 * Supply lines from Poland to L'viv and West Ukraine.

Increased NATO presence in Poland

 * NATO forces in Poland

At the 2022 NATO Madrid summit, U.S. President Joe Biden announced that a new permanent U.S. military base will be established in Poland - the 5th Army's Headquarters. The 5th Army has had a forward command post in the Poznań since last year. Poland's government has long requested such a base. (Reuters) (Notes from Poland) (DoD)

Elementary properties
For any vector $$x,$$ the scalar $$\langle x, x \rangle$$ is always a real number, $$\langle x, \mathbf{0} \rangle = \langle \mathbf{0}, x \rangle = 0,$$ and $$\langle x, x \rangle = 0$$ if and only if $$x = \mathbf{0}.$$

Inner products are also (sometimes called, ) in their second argument, meaning that any vectors $$x, y, z$$ and any scalar $$s,$$ $$\langle x, y + z \rangle = \langle x, y \rangle + \langle x, z \rangle$$ $$\langle x, s y \rangle = \overline{s} \langle x, y \rangle$$ Linearity in one argument and antilinearity in the other are the defining properties of a and complex inner products can equivalently be defined as being positive-definite sesquilinear form.

Linearity in one argument and Hermitian symmetry are the defining properties of a, which is a special type of sesquilinear form. A complex sesquilinear form is Hermitian if and only if $$\langle x, x \rangle$$ is real for all $$x,$$ where the latter condition is guaranteed by Hermitian symmetry. Consequently, an inner product is sometimes equivalently defined as being a.

Hermitian symmetry holds if and only if the real part of $$\langle \cdot, \cdot \rangle$$ is a symmetric map and its imaginary part is an antisymmetric map. So in the case of $$\mathbf{F} = \R,$$ Hermitian symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Consequently, a inner product (but not a complex inner product) can equivalently defined as being a.

Convention variant
Omitted

Completeness inner product spaces
Every inner product space $$(V, \langle \cdot, \cdot \rangle)$$ becomes a normed space when it is endowed with the canonical norm induced by its inner product, which is $$\|x\| = \sqrt{\langle x, x \rangle}.$$ An inner product space is called a if this normed space is a Banach space. Every inner product space $$(V, \langle \cdot, \cdot \rangle)$$ can be linearly isometrically embedded as a dense vector subspace of some Hilbert space $$\left(H, \langle \cdot, \cdot \rangle_H\right)$$ that is called its ; its inner product is the unique continuous extension $$\langle \cdot, \cdot \rangle_H : H \times H \to \mathbf{F}$$ of the original inner product $$\langle \cdot, \cdot \rangle : V \times V \to \mathbf{F}.$$ Hausdorff completions are unique up to an isometric linear isomorphism.

Some examples
Omitted

Continuous functions on an interval

 * ORIGINAL TITLE OF THIS SUBSECTION WAS "Hilbert space". I also removed the first sentence from this subsection (given here ). Nothing else in this subsection was changed.

An example of an inner product space which induces an incomplete metric is the space $$C([a, b])$$ of continuous complex valued functions $$f$$ and $$g$$ on the interval $$[a, b].$$ The inner product is $$\langle f, g \rangle = \int_a^b f(t) \overline{g(t)} \, \mathrm{d}t.$$ This space is not complete; consider for example, for the interval $[−1, 1]$ the sequence of continuous "step" functions, $$\{ f_k \}_k,$$ defined by: $$f_k(t) = \begin{cases} 0 & t \in [-1, 0] \\ 1 & t \in \left[\tfrac{1}{k}, 1\right] \\ kt & t \in \left(0, \tfrac{1}{k}\right) \end{cases}$$

This sequence is a Cauchy sequence for the norm induced by the preceding inner product, which does not converge to a function.

Random variables
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Basic results, terminology, and definitions
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Real and complex parts of inner products

 * THIS SUBSECTION NEEDS MORE WORK


 * OMITTED

Orthonormal sequences
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Operators on inner product spaces
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Related products
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