Properties of Semi-Inner Product
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Theorem
Let $V$ be a vector space over $\Bbb F \in \set {\R, \C}$.
Let $\innerprod \cdot \cdot$ be a semi-inner product on $V$.
Denote, for $x \in V$, $\norm x := \innerprod x x^{1 / 2}$.
Then, $\forall x, y \in V, a \in \Bbb F$:
- $(1): \quad \norm {x + y} \le \norm x + \norm y$
- $(2): \quad \norm {a x} = \size a \norm x$
Proof
Proof of $(1)$
For $x, y \in V$, compute:
| \(\ds \norm {x + y}^2\) | \(=\) | \(\ds \innerprod {x + y} {x + y}\) | Definition of $\norm \cdot$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \innerprod x x + \innerprod x y + \innerprod y x + \innerprod y y\) | Linearity of $\innerprod \cdot \cdot$ | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \innerprod x x + \sqrt {\innerprod x x \innerprod y y} + \sqrt {\innerprod y y \innerprod x x} + \innerprod y y\) | Cauchy-Bunyakovsky-Schwarz Inequality for Inner Product Spaces | |||||||||||
| \(\ds \) | \(=\) | \(\ds \norm x^2 + 2 \norm x \norm y + \norm y^2\) | Definition of $\norm \cdot$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\norm x + \norm y}^2\) |
Taking square roots on either side gives the result.
$\Box$
Proof of $(2)$
For $x \in V$, $a \in \Bbb F$, compute:
| \(\ds \norm {a x}^2\) | \(=\) | \(\ds \innerprod {a x} {a x}\) | Definition of $\norm \cdot$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds a \innerprod x {a x}\) | Linearity of $\innerprod \cdot \cdot$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds a \overline {\innerprod {a x} x}\) | Conjugate symmetry of $\innerprod \cdot \cdot$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds a \overline a \overline {\innerprod x x}\) | Linearity of $\innerprod \cdot \cdot$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \size a^2 \norm x^2\) | Modulus in Terms of Conjugate |
Taking square roots on either side gives the result.
$\blacksquare$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Corollary $1.5$