Inner product space and subspaces
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Exercise:
Let V be an inner product space possibly of infinite dimension and Usubseteq V a finite dimensional subspace. Then VUoplus U^perp.
Solution:
Proof. By the Lemma before we have Ucap U^perp. So it's enought to show that U+U^perpV. Put r:textdimU and pick an orthonormal basis e_...e_r is not the standard basis or something like that for U. Let vin V. Define tildeP_Uv:langle v e_rangle e_+...+langle v e_rrangle e_r in U. The orthogonal projection of v to/on U. We have vtildeP_Uv+v-tildeP_Uv *. We claim that v-tildeP_Uvin U^perp. Indeed langle v-tildeP_Uv e_irangle langle v e_irangle - _i^r langle v e_krangle langle e_k e_irangle langle v e_irangle - langle v e_i rangle . This holds forall i hence v-tildeP_Uvperp textSpe_...e_rU i.e. v-tildeP_Uvin U^perp. Going back to * we see that vtildeP_Uv+v-tildeP_Uv. This holds forall vin V Longrightarrow VU+U^perp. Remark: If Usubseteq V is a finite dimensional subspace then U^perp is a complement of U.
Let V be an inner product space possibly of infinite dimension and Usubseteq V a finite dimensional subspace. Then VUoplus U^perp.
Solution:
Proof. By the Lemma before we have Ucap U^perp. So it's enought to show that U+U^perpV. Put r:textdimU and pick an orthonormal basis e_...e_r is not the standard basis or something like that for U. Let vin V. Define tildeP_Uv:langle v e_rangle e_+...+langle v e_rrangle e_r in U. The orthogonal projection of v to/on U. We have vtildeP_Uv+v-tildeP_Uv *. We claim that v-tildeP_Uvin U^perp. Indeed langle v-tildeP_Uv e_irangle langle v e_irangle - _i^r langle v e_krangle langle e_k e_irangle langle v e_irangle - langle v e_i rangle . This holds forall i hence v-tildeP_Uvperp textSpe_...e_rU i.e. v-tildeP_Uvin U^perp. Going back to * we see that vtildeP_Uv+v-tildeP_Uv. This holds forall vin V Longrightarrow VU+U^perp. Remark: If Usubseteq V is a finite dimensional subspace then U^perp is a complement of U.