Semi-Inner Product with Zero Vector

Theorem

Let $\struct {V, \innerprod \cdot \cdot}$ be a semi-inner product space.

Let $\mathbf 0_V$ be the zero vector of $V$.

Then for all $v \in V$:

$\innerprod {\mathbf 0_V} v = \innerprod v {\mathbf 0_V} = 0$

Proof

\(\ds \innerprod {\mathbf 0_V} v\)

\(=\)

\(\ds \innerprod {0 \cdot \mathbf 0_V} v\)

\(\ds \)

\(=\)

\(\ds 0 \cdot \innerprod {\mathbf 0_V} v\)

Semi-Inner Product Axioms: Sesquilinearity

\(\ds \)

\(=\)

\(\ds 0\)

$\Box$

\(\ds \innerprod v {\mathbf 0_V}\)

\(=\)

\(\ds \overline {\innerprod {\mathbf 0_V} v}\)

Semi-Inner Product Axioms: Conjugate Symmetry

\(\ds \)

\(=\)

\(\ds \overline 0\)

From above

\(\ds \)

\(=\)

\(\ds 0\)

$\blacksquare$

Also see

• Inner Product with Zero Vector

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Definition $1.1$