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Mathematics for Physics

232 P

A parameter that is a real number at least one that is used to define a family of norms on sequences and on functions by choosing the power in a sum or integral.

definition [d] (\(p\) = Exponent Parameter) A fixed real number \(p \geq 1\): the power to which absolute values of sequence terms or of function values are raised in forming the \(\ell^{p}\) sum or the \(L^{p}\) integral, before the outer \(\dfrac{1}{p}\) root that defines the \(p\)-norm.

where

  • \(p\) is the exponent parameter.
  • \(\ell^{p}\) is the space of \(p\)-summable sequences.
  • \(L^{p}\) is the corresponding Banach space of \(p\)-integrable functions.
  • the \(p\)-norm of a sequence \(x = (\xi_{j})\) is \(\lVert x \rVert_{p} = \bigl(\sum_{j}|\xi_{j}|^{p}\bigr)^{\dfrac{1}{p}}\).

Note:

  • Kreyszig introduces \(p\) with the phrase “let \(p \geq 1\) be a fixed real number.”
  • for \(p > 1\), the companion \(q\) defined by \(\dfrac{1}{p} + \dfrac{1}{q} = 1\) is the conjugate exponent.

definition [d] (\(p\) = Norm Family Parameter) A real parameter in the range \(1 \leq p < \infty\) that labels the one-parameter family of \(p\)-norms \(\lVert\,\cdot\,\rVert_{p}\) on sequence spaces and on spaces of continuous functions.

where

  • \(p\) is the parameter of the norm family.
  • \(\lVert x \rVert_{p} = \bigl(\sum_{j}|x_{j}|^{p}\bigr)^{\dfrac{1}{p}}\) on \(\ell^{p}\).
  • \(\lVert f \rVert_{p} = \bigl(\int |f|^{p}\, ds\bigr)^{\dfrac{1}{p}}\) on suitable function spaces.

Note:

  • as \(p \to \infty\), \(\lVert\,\cdot\,\rVert_{p}\) tends to the supremum norm \(\lVert\,\cdot\,\rVert_{\infty}\).
  • the case \(p = 2\) recovers the Euclidean and Hilbert norms associated with an inner product.

definition [d] (\(p\) in Mathematical Physics) A positive integer appearing in the finite-dimensional \(p\)-norm on \(\mathbb{C}^{n}\),

  • \(\lVert a \rVert_{p} \equiv \left( \displaystyle\sum_{i=1}^{n} |\alpha_{i}|^{p} \right)^{\dfrac{1}{p}}\) ,

and, in the infinite-dimensional setting of square-integrable functions, the special value \(p = 2\) that is the power in the integrability condition defining \(L^{2}_{w}(a,b)\).

where

  • \(a = \{\alpha_{i}\}\) is a vector in \(\mathbb{C}^{n}\).
  • \(p\) is a positive integer in the finite-dimensional formula.
  • \(L^{2}_{w}(a,b)\) is the weighted space of square-integrable functions on \([a,b]\).
  • the superscript \(2\) in \(L^{2}\) is this power \(p = 2\).

Note:

  • Hassani’s infinite-dimensional development focuses on the Hilbert case \(p = 2\).
  • Arfken likewise works primarily with square-integrable functions rather than general \(L^{p}\).

232.1 Historical Notes

Riesz introduced the parameter \(p\) in 1910. He replaced the assumption of quadratic integrability by the integrability of \(|f|^{p}\). Each number \(p\) greater than \(1\) determines a function class \(L^{p}\). The letter \(p\) names that power in \(|f|^{p}\).

232.2 Elementary Example

232.2.1 Simple

The parameter \(p\) is a real number at least \(1\). The choice \(p = 1\) gives the sum of absolute values.

\[ p = 1 \]

\[ \lVert x \rVert_{1} = |\xi_{1}| + |\xi_{2}| + |\xi_{3}| \]

\[ x = (\xi_{1},\xi_{2},\xi_{3}) = (1,-2,3),\quad \lVert x \rVert_{1} = 6 \]

where

  • \(p\) is the exponent parameter.
  • \(\lVert x \rVert_{p}\) is the \(p\)-norm.

232.2.2 General

For \(p = 2\) the same three-term sequence uses squares under a square root.

\[ p = 2 \]

\[ \lVert x \rVert_{2} = \bigl(|\xi_{1}|^{2} + |\xi_{2}|^{2} + |\xi_{3}|^{2}\bigr)^{1/2} \]

\[ x = (1,-2,3),\quad \lVert x \rVert_{2} = \sqrt{14} \]

where

  • \(p \geq 1\) labels the family of \(p\)-norms.

232.3 References

  1. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — fixed \(p\geq 1\) in \(\ell^{p}\) and \(L^{p}\); conjugate exponents \(\dfrac{1}{p} + \dfrac{1}{q}=1\).
  2. Gamelin, T. W., & Greene, R. E. Introduction to Topology, 2nd ed. Dover, 1999. — one-parameter family of \(p\)-norms for \(1\leq p<\infty\).
  3. Hassani, S. Mathematical Physics, 2nd ed. Springer. — \(p\)-norm on \(\mathbb{C}^{n}\); \(L^{2}_{w}\) with power \(2\).
  4. Pietsch, A. History of Banach Spaces and Linear Operators. Birkhäuser, 2007. — Riesz and the introduction of \(L^{p}\) via the power \(p\) in \(|f|^{p}\).